Zuko's adorable face + hilarious failed attempts at being social = this trope. Zuko's adorkableness started in season 2 in "Ba Sing Se", particularly his date in Tales of Ba Sing Se. Exhibit A.◊ His Fan Nicknames are "Zudork", "Dorko", and "Failko", and that encapsulates his attempts at being normal. Poor guy reallyCannot Tell a Joke. It's rather funny and odd because he spends most of the time being a no-nonsense warrior, Wise Prince and stern pursuer. But when he's forced to be social, he fails hilariously.
Zuko:(after months of trying to capture or kill Aang, wishing to now join the group and teach Aang firebending) Hello, Zuko here. But I guess you probably already know me, sort of. Uh, so, the thing is I have a lot of Firebending experience and I'm considered to be pretty good at it. Well you've seen me... you know when I was attacking you? Uh, yeah, I guess I should apologize for that, but anyway I'm good now. I mean, I thought I was good before but now I realize I was bad, but anyway... I think it's time I joined your group and taught the Avatar Firebending...
The Earth King who is extremely naive for his age since he was kept in the palace his entire life while Evil Chancellor Long Feng took care of politics. He is sometimes even called "King Adorkable." Seriously, his pet bear is much more worldly and mature than he is.
Meng from "The Fortune Teller" is a female example. A young girl with pigtails that defy gravity and a missing front tooth, who believes Aang is her destined lover solely because of his big ears.
The professor/archaologist Professor Zei from "The Library." Loves his work with a passion, fanboys over Aang not because he's the Avatar but because he is a "living fossil," tries to communicate with Appa (while hushing Momo), admires Won Shi Tong's "foxy assistants" (actual foxes) for being handsome and clever, and can not understand what is so funny about the term "flying buttresses" because he's too busy being genuinely enthusiastic about architecture. He's also almost a Perpetual Smiler in a non-creepy way. Yeah, he lives in his own little world.
Sokka, on occasion. His moments of over-enthusiasm are the farthest thing from cool, but still endearing in their way.
Azula of all people has her moments, as her tendency to be perpetually grim and morbid make her extremely socially awkward. Especially evident in "The Beach", where she tries to act normal and socialize at a party.
Azula: That's a sharp outfit, Chan. Careful, you could puncture the hull of an empire-class Fire Nation battleship, leaving thousands to drown at sea.
Most of the science types are your classic hot badass types... then you have Hank Pym, who will geekily squee about the science-influenced powers of a guy trying to kill him.
On the female side of things, there's Wasp, who also happens to be Hank's not-girlfriend. Seriously, she's d'awwww-worthy in most of her scenes.
Batman: The Animated Series had quite a few. Most of the rogues can fall into this category at some point in their portrayal, and characters like Alfred have their moments. The series is simply full of it. Harley Quinn is probably most notable for being painfully adorkable when she tries to interact with the public after her release from Arkham (short lived), but also good examples are the Ventriloquist, when he's simply being Arnold, and Jervis Tetch, when he's either at war with himself, or prior to his spiral into madness. Tetch in particular, had some incredibly adorable moments of complete social ineptitude in his office; talking to people, especially pretty people was not his forte.
Lola from Charlie and Lola. She's eccentric, clumsy, and at times rather annoying - but you just gotta love her.
Jérémie from Code Lyoko. He's got a 99.87 grade average and ungodly computer programming skills, and has worked tirelessly for two years to give Aelita a normal, stable life. So that makes him a colossal nerd, an angel of a love interest, and just downright adorable physically.
Danny, especially in his human form. Sweet, awkward, shy, clumsy, those wide baby blue eyes, and more. His phantom form in the earlier episodes could count, being nervous and scared when he started out.
Daria, while she's not shy, she's unpopular due to her passive, anti-social and snarky attitude and it somehow makes her more relatable and endearing as a character despite her emotionless personality.
Ed. He's a lovable oaf with a silly, playful and innocent personality, and has a number of odd obsessions including sci-fi movies and comic books.
Edd. Everything about him is adorkable. He's cute, geeky, neurotic, endearingly shy, kind, polite, and quite a Shrinking Violet, especially around girls. And that's why all the girls (In-Universe and out) find him adorable.
Jimmy being a cute, delicate, accident-prone kid who cries easily and has a lot of Woobie moments.
Jonny, a weird but good-natured Cloudcuckoolander. He's especially adorkable when he's interacting with his Companion Cube Plank, a piece of wood who is also his best friend.
Meg Griffin because she looks like a geek (she wears glasses), she's called a "wannabee loser", and because of her Butt Monkey status she's constantly picked on and made fun of. You can't help finding her downright Adorkable.
A parody of Hugh Grant, poking fun on his many roles and typecasting: "I'm just so charmingly befuddled."
Phillip J. Fry from Futurama. Mostly due to his childish nature and his kooky mannerisms, and he's a Trekkie. He's not seen as particularly attractive in-universe, but he manages to date some fairly hot women at times.
Tenzin, which is especially funny because he spends most of his appearances being super serious and pretending to not have a sense of humor. And then Korra brings up his romantic past with Lin Beifong. Tenzin's Not So Stoic/The Comically Serious moments are largely what contribute to his Adorkable status. A lot of his facial expressions during his Not So Stoic moments wouldn't exactly look out of place on his dad.
Lilo. She's such a friendly, sweet-natured, but socially awkward girl who has a strange obsession with Elvis and taking photos of fat people, and loves hanging out with her best friend, Stitch, an alien from outer space.
Twilight Sparkle is essentially a former shut-in who's just learning how to maintain friendships. Despite this she can still spend days cooped up in her home reading books, and has evinced a tendency toward Super OCD that borders on the scary. It's on full display when we see her dancing skills (or lack thereof) at her birthday party in Canterlot, and she takes it Up to Eleven when she was a filly, in "The Cutie Mark Chronicles".
Twilight's brother Shining Armor was a huge nerd before becoming a guard. When he gets a chance to talk to his crush Cadance for the first time, he manages to croak out a very awkward "Hello"; Cadance's observation is "Could he be any more adorable?"
Rainbow Dash, Passionate Sports Girl and Lovable Jock, can be adorkable, particularly when she's fangirling over The Wonderbolts and Daring Do, an Indiana Jones Expy. She's even willing to wear a very dorky looking sweater while posing with an owl. She also has some really cute moments with Fluttershy like going to see the Butterfly Migration just because Fluttershy asked.
In addition to being the regular kind of adorable, Fluttershy also has her share of adorkable traits as well. Normally she's very well... shy, but if something piques her interest, such as animals, she'll get excited and starts talking your ear off.
Mikey's naivety mixed with his Large Ham tendencies make him adorably dorky.
The Pale Kids are a group of nerdy ten year olds who spend their recesses playing Daggers & Dragons and reading comic books.
Miss Grotke has many cute, dorky moments throughout the series. It's her over-enthusiasm about everything that really hits it.
T.J. can be pretty adorkable. His ditzy Book Dumb moments, his overly enthusiastic and optimistic nature, and he's got an enormous heart. And he was a Shrinking Violet in kindergarten! And he's been revealed to be a huge geek when it comes to comics.
Gus is a shy cutie who doubles as The Woobie. He's a timid, naive, insecure yet good natured boy.
The title character from Skippy: Adventures in Bushtown shows shades of this, but Pos the cameraman possum probably fits the trope even better. In the Title Sequence, Pos is shown trying to film Skippy walking towards him but the he trips walking backwards and falls on his butt so Skippy doesn't bump into him. It's only 1 or 2 seconds long and yet it is completely adorable.
While not all iterations count, Donatello tends to fall into this category, particularly the 2003 show's version. Soft spoken and gentle, noticeably awkward and even clumsy compared to his brothers, and that's not even getting into the Teen Genius bit. Considering that the turtles don't have irises most of the time, the amount of emotion he conveys through his eyes alone is incredible, and can make big-hearted viewers want to hug him.
The 2012 show has Leonardo take the steed. While still the leader of the group, he also has a fondness for an old Sci-fi show and uses this as his basis of being a leader. And it is adorable. It's approaching ridiculous moe levels combined with his big blue eyes and the expressions he makes.
While this term is used to describe Ho Yay between Noah/Cody, the Paris episode proves that Noah on his own certainly counts.
Beth has many traits of adorkability, especially how awkward she acts when she interacts with the other characters.
Winx Club: Tecna acts like a robot a lot and it's suspected she didn't have any friends or feel emotions before meeting the rest of the Club, as well as her boyfriend, Timmy. No bowtie, but still adorkable, as the pairing is.
Tim Drake as the new Robin in season 2 has his moments, especially when he gets nervous and flustered when put in charge of leading team alpha and his facial expressions to his teammates' antics.
Captain Marvel is this almost from his first appearance, where he clearly just wants to hang out with the Team and be their friend, trying to do everything he can to be nice, friendly and helpful to them. However, since the kids are generally mistrustful of the League (as some of their mentors have been keeping secrets from them), they wind up keeping him at arms length until they find out that he's just a 10 year old kid whose superpowered form happens to look like a 30 something year adult.