Friday from Robinson Crusoe may be considered a Trope Codifier. He is a Caribbean tribesman rescued from cannibals by the eponymous castaway. Friday's English is basically a broken pidgin, which never, ever improves (he's a non-white servant, after all), even after being with Crusoe for years and living in England for a time (as established in the little-known sequel). An example from a dialogue between Crusoe and Friday:
Master. - How beat? If your nation beat them, how came you to be taken?
Friday. - They more many than my nation, in the place where me was; they take one, two, three, and me: my nation over-beat them in the yonder place, where me no was; there my nation take one, two, great thousand.
Seemingly averted in J. R. R. Tolkien's The Lord of the Rings. While the orcs' lines appear to be perfectly good, clean English, the narrator reveals that the orcs actually utter such repulsive profanity in such a degraded gibberish that he feels no need to bother reproducing it, instead merely paraphrasing them. To a philologist like Tolkien, using language so improperly was a clear sign of how degenerate and inferior they were.
One character speaking sub-fluently is Ghan-buri-Ghan, though it's Justified as his people have no real contact with Westron-speaking peoples, so it is not strange that he barely speaks it. Also, his people are actually rather smart and wise, and even the characters are a bit surprised at what they thought to be stupid primitives.
Subverted in the Phule's Company novels. Tuskanini, one of the Legionnaires of the titular military company speaks, rather brokenly, the English equivalent of the series. However, he is the company clerk, incredibly intelligent, capable of reading 10-15 books in a night, and plans to become a teacher. He speaks it brokenly because he learned the language manually, and chooses not to rely on a translator, despite the presence of normally functioning translation devices.
To be clear, said character is not human, but speaks a human language (if somewhat badly). He's essentially a warthog minotaur and half his problem is that his mouth is the wrong shape.
Animorphs has the Hork-Bajir, who talk in very broken English with some of their own language thrown in. It's said by many characters that Hork-Bajir are "not the geniuses of the galaxy," but in what appears to be an unintended subversion, the books that feature them heavily have shown them to demonstrate street smarts that exceed their linguistic skills.
Later revealed that the trope is justified. The Hork-Bajir's intelligence and language skills were being kept down and manipulated by an ancient conspiracy on their own planet. An alien race originally created them for ecological purposes and actively worked to keep them as dumb as possible so they wouldn't discover the truth. The low language skills were intentional as it prevented smarter members of the race from conveying complex concepts they had grasped to the slower members, for example a Hork-Bajir genius has discovered art but his language lacked the ability to represent abstract elements. When he said a etching was of a friend, others heard that he was insisting the markers were the friend, which lead them to believe he was crazy.
Anthony Boucher's science-fiction story Barrier presents a future in which this has been done deliberately: only four languages remain extant, and all of them have been "regularized": there are no longer any irregular verbs ("is" becomes "bees"), all plurals are formed by adding s or es ("men" is now "mans"), articles have been dropped completely, and so forth. It sounds odd, but in fact probably would be considerably easier to learn.
In C. J. Cherryh's Chanur Saga, the Mahendo'sat, though no more nor less intelligent than the other oxygen-breathing space-faring species, have a difficult time learning the languages of other species, so when speaking to other species their merchants use a pidgin language which is rendered in English in a "primitive" sounding manner.
They speak it among themselves as well, from having hundreds or thousands of different dialects on their home planet, sort of like Chinese, with the differences between Mandarin and Cantonese and such. Because it's simplified so much, it's used as the de facto trade language between the various aliens.
Lampshaded to a degree by the titular character of the Firekeeper series, who was Raised by Wolves. She insists that using more words than she needs to make her point is pointless. Also Subverted to a degree in that when speaking with animals (which is her native "language") she is depicted as having normal grammatical skills.
In the Star Trek novel The Galactic Whirlpool, a culture stuck on a lost Generation starship for three centuries develops a dialect of English that No Takes Candle. They still know full English and use it in religious ceremonies, but consider it stilted and overly ornate for everyday use.
Turtle Heart is to be surprised he is not to be mentioned yet. But Turtle Heart is to have been a small part that is to be served purely to be questioning Nessarose's father.
The Party was deliberately imposing this trope on the people of Oceania in 1984. New Speak was an effort to chop down the English language and strip away words for concepts (like love and rebellion) which were dangerous to the leaders, under the guise of efficiency.
C. S. Lewis' Out of the Silent Planet inverts this—the supposedly civilized scientists who intend to conquer Malacandra don't bother much with the local lingo, and as a result sound crude and vicious next to the linguist they've brought as a hostage.
Various species of Redwall use different varieties of English, mostly based on actual British accents, but a few fall into this trope. The Sparra inexplicably use what seems to be old-stereotype Native American accents ("Can you imagine Friar Hugo's face when Warbeak tells him to 'burn fishworm good'?"), and some of the vermin use very broken English ("Dis de blade wot stop your breath"). It doesn't seem to be a sign of stupidity in the case of the Sparra, though, just that they have very little contact with the mammals.
Riverworld: Mark Twain's caveman buddy Kazz speaks in heavily accented, broken English. He's smart enough, but his vocal tract is not sufficiently evolved toward speech. Flashbacks have to be told in a generic style.
In the Xanth novels by Piers Anthony, Ogres supposedly speak in broken-English couplets. However, as evidenced by more than one book, if one abandons their prejudices, they can hear the ogre as he actually is speaking, in complete sentences. Now, Ogres pride themselves on being both ugly and stupid, but seriously.
In Manifold: Origin, Neanderthals speak more English, but the grammar is still broken. The Daemons (no relation) hear something similar when humans try to speak their language, though it's poorly represented since Most Writers Are Human.
In David Sedaris' essay collection Me Talk Pretty One Day, the essays detailing his attempts to learn French contain many examples of the translated English of his horribly mangled French. For example, when attempting to ask a butcher if those are indeed cow's brains, he asks "Is thems the thoughts of cows?"
Lakota Indian Nannie Little Rose talks like this in the book My Heart Is on the Ground, which is supposed to be her diary as she goes to Carlisle Indian Industrial school, an (actual) school meant to teach Native Americans how to be "white" (no, really). * The school founder's motto was "Kill the Indian to save the man!" Then, as if to make up for this, she learns fluent English in ten months of being there.
Used in Watership Down when the animals speak Hedgerow, a kind of inter-species pidgin, which appears to owe a lot to Italian for whatever reason. Especially played up with mice ("You want-a nice grass? Plenty-a nice-a grass!") and Kehaar the gull.
The resemblance of the translated "hedgerow" pidgin to the stereotype of the English speaker who thinks that adding "-a" or "-o" to English words to make them understandable to Italians or foreigners generally is almost certainly intentional. It is meant to show the generally condescending attitude of rabbits to mice and other smaller animals.
In John Steinbeck's East of Eden, the Chinese servant speaks "Chinee" until a white man observes how very odd it is that no one Chinese ever speaks good English, whereupon he reveals it's intentional, for those who expect it. He was in fact born in the United States and has lived his entire life there. He only reveals his true fluency and personality to people he trusts. He switches to standard English with his employer while the employer is suffering Heroic BSOD.
In E. E. “Doc” Smith's Skylark Series, a Japanese servant speaks pidgin. In the second book, his employer speaks of it, and the servant says he started to learn English too late, and it's too different from his native tongue. (Then the employer invents a gadget to allow people to transfer linguistic knowledge.)
Mannie's narration in The Moon Is a Harsh Mistress comes off like this, as it's in Lunar English. He reveals in the narrative that he can in fact speak standard English, but it's not what he prefers. And anyone who says differently is a yammerhead.
Consider that his narration bears a certain similarity to Russian, most notably the omission of the definite article, which neither Russian nor Chinese (among a variety of other languages) have. A smattering of other Russian-inspired or -rooted words and phrases appear throughout the book, unsurprising considering that apparently many of the first colonists of Luna were Russian prisoners. Think of it as a sort of heavily watered-down Nadsat.
The Coyote Dialect as we hear it in the Hank the Cowdog series. Hank talks about it as if it's an actual language, but it's never clarified whether or not we're just hearing a translation.
Discussed in The Bonesetter's Daughter, in which the main character is concerned about her mother Lu Ling being misdiagnosed in a dementia test because of her poor English and the fact that she usually translated things people said to Chinese in her head, then responded in English.
Happens several times in Harry TurtledoveWorld War books with the Race, reptilian conquerors to invade Earth during World War II, although subverted in that they are a highly-advanced species. Their representatives learn major human languages, but tend to speak in this manner (e.g. "maybe you help us now"), often trying to find proper equivalent in the given human language for a specific word with the typical "how you say". This differs from lizard to lizard, though, and some get better as the series progresses. Given their physiology, though, their speech is also peppered with Sssssnake Talk. It also happens, as a necessity, between Liu Han and Bobby Fiore, as neither initially knows the other's language. They eventually develop a mix language of sorts, a mishmash of Chinese, English, Race, and sign. Only the two of them can understand it. The books show this as a You No Take Candle-like speech.
Ayla in the Earth's Children books tends to do this when she's learning a new language.
This backfired on Alexandra (Zan) Ford in the YA novel Saturday The Twelfth Of October. Thrown back into prehistoric times by a convenient glitch in the space-time continuum, Zan is discovered by two cave people about her own age. She goes into the "me Tarzan, you Jane" routine and says "Me Zan". For the rest of the book, the cave people call her Meezzan. She even starts thinking of herself as Meezzan during the year or so she lives with them.
Tiger Lily and her tribe in Peter Pan. Oddly enough, they use an Asian Speekee Engrish accent, despite being Indians. (Perhaps justified in that everything in Never Land is based on children's imagination, and children rarely do the research.)
A fairly realistic one is done in Harry Turtledove's Supervolcano: Eruption with a Filipina store clerk, whose English is understandable but displays some grammatical problems that actually do tend to happen to many Filipinos in Real Life. However, it gets ridiculous when a police officer has to mime out the word "mask" to get her to understand. English is common enough in the Philippines that many English-language shows and books are left untranslated, and the word maskara (a localized spelling of the Spanish word mascara) is found in the major Filipino languages and dialects. She should have had no problem understanding "mask."
In The Acharnians by Aristophanes, the Persians at the assembly in Act One (and later on) are depicted speaking butchered Greek, making this trope Older Than Feudalism. Translations often attempt to keep the comedic nature of the depiction by applying You No Take Candle in the translation as well. One notable English version translates one line as "No getty goldy, wide-assed Athenian."
In Galax-Arena by Gillian Rubinstein, the peb - children abducted from various third-world countries around the world, now imprisoned on a hostile alien planet - have developed a form of pidgin English they call patwa, which serves as the only common language and means of communication between them all.
In Primo Levi's Auschwitz memoir If This Is a Man, this trope occurs frequently, owing to the communication difficulties inherent in living in close proximity to a group of strangers from all the nations of Europe. If two prisoners have no other language in common, they most often converse in butchered German, that being the language they all have rudimentary knowledge of by necessity. For example, one Pole says to Levi, "Du Jude, kaputt. Du schnell Krematorium fertig." (You Jew, finished. You soon ready for crematorium.)