These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Badass Decay: Kate's arguably suffering this in the current run following the Annual. Ever since she went solo, she's had trouble not getting beat up during fights. Somewhat justified by the fact this is the first time she's gone solo and is now in a fragile place, she's trying out a new thing, and so far her fights have been her running into the fight unprepared and/or caught off-guard. Then there's the fact that each of the people she's been beaten up by were in league with Madame Masque, when most of her experience with mooks have been the highly untrained tracksuit bros.
Broken Base: While generally well-received at first, his ongoing has resulted in a four-way split. There's people who've grown tired of Clint and are only reading for Kate's current solo chapters, another who're the opposite and dislike her for hogging the spotlight, then there's those who like both and, in spite of the series flaws, are enjoying the current comics, and then those who dislike everything or at least enough to drop it.
Ensemble Darkhorse: Lucky, or Pizza Dog, as everyone calls him. In fact, issue #11 of the current series is told from his perspective as he tries to solve a murder on his own.
The Avengers has significantly boosted the support for Clint and Black Widow, but this is mostly for the film universe, where neither have any love interests yet (and given that an overwhelming majority of the film's viewers are unfamiliar with the comics, there's far less alternatives). Despite that, Bobbi has popped up in MCU-set fics as a love interest for Hawkeye, either for this trope, or for another, more sinisterset of tropes.
Fanon Discontinuity: Clint's entire relationship with Jessica is this among many fans who felt that it came out of left field. Some people acknowledge the relationship but say that it ended before Hawkeye.
Hawkward! (Used as a result of the many, many, uncomfortable romantic situations Clint's been involved in.)
The "Hawkblock", using Clint's face in his old costume to cover his crotch in one issue, was so popular it became a T-shirt.
The Scrappy: Kate, but only in some circles. Others say she's the saving grace of the series, see Broken Base above.
The Woobie: Oh, boy. Abused as a child, constantly screwed over by the people he cared about, tried to become a hero and was branded a criminal. He was later killed by the Scarlet Witch, but resurrected in House of M and had his memories restored, which meant that he knew that he had died, and stayed alive even after Wanda changed almost everything back. Mockingbird, who had been killed, was found alive during Secret Invasion, but it turned out that she was a skrull. The real Mockingbird also turned out to be alive, but they still ended up getting divorced.
Cemented in the Hawkeye and Mockingbird short lived ongoing, where a subplot is that she's in a dark place right now, and there's nothing he can do to get her out of it and it's killing him that he can't help her, all the while being around her is pulling him into a bad place too. She counts too, due to the aforementioned death/skrull abduction and the related angst, her being raped by Phantom Rider, the dark places she went during the series, and the fact her mom gets nearly killed because of her and her brother hates her guts for faking her death.
Iron Woobie: Both of them. It's why they work together.
What Could Have Been: At one point Stan Lee reportedly seriously considered having it be revealed that Hawkeye was actually Captain America's biological son, since they already had a strained surrogate father/son thing going on, but the idea never came to fruition. And probably better that it didn't, what with Marvel's sliding time scale (it would have been stretching credibility by the 80s, by the 90s it would have probably been retconned as Hawkeye being Cap's grandson).
What Measure Is a Non-Badass?: Despite probably being one of the Avengers longest lasting members, and the longest lasting, non-super genius Badass Normal member, Clint has struggled to get the same kind of respect Captain America or Iron Man get outside of universe.