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: Ever since Kate went solo, she's had far less success fighting mooks, probably as a result of Masque launching personal attacks against her shaking her confidence, combined with the situations she keeps catching herself in.
Broken Base: While generally well-received at first, his ongoing has resulted in a 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 think the arguments are stupid.
In particular, the Badass Decay noted above; it is true Kate's success rate has gone down, but it has been pointed out that, while a Badass Normal, she's not typically a One-Man Army, and isn't used to fighting by herself. Some consider it Badass Decay, others consider it Reality Ensues. Should be noted that it does make for a comparison between her and Clint: He too doesn't get out of fights without some bruises, despite his greater experience and comparatively less-trained mooks. If she was taking down everyone without effort while he was struggling, it would probably lead to accusations of her being a Creator's Pet.
The announcement that the run is ending, and will be followed by Jeff Lemire taking over the title, has also been contentious in two ways. For some, he has big shoes to fill and so he runs the risk of making a Contested Sequel, while others are optimistic and/or just want to see someone other than Matt Fraction writing the book. On the other hand, Lemire is most recently known for writing Green Arrow, and helped save the title from cancellation, so many who enjoyed Fraction's work are glad that they picked him to follow up, while others, who either didn't enjoy his treatment for Green Arrow or didn't read it, are still just as weary.
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.
Almost everything Brian Bendis does with Clint whenever he uses him too; Bendis has generally recieved criticism for his use of Clint generally writing him as a one-note jackass while ignoring a lot of his core character traits (most notably, his no-killing policy, as Bendis generally has him itching to go for the kill shot). Because of that, even people who like Bendis' Avengers stuff generally don't enjoy his writing of 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.
Older Than They Think: A lot of people tend to accuse his relationship and marriage to Mockingbird as an attempt by Marvel to rip-off DC's Green Arrow and Black Canary relationship. This is despite the fact that Arrow and Canary only got married during the 2000s, while Clint and Bobbi got married almost as soon as they met, and as such, while the two couples have many similarities, they ultimately, and largely, had a very different relationship.
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.
He gets it especially bad in the Marvel Cinematic Universe, though mostly from critics rather than the actual fanbase. Basically, since he couldn't take Loki on in a fight (bear in mind, Loki caught him by surprise), got brainwashed, and was something of a Distressed Dude for Black Widow to save, combined with not having superpowers and running out of arrows in the final fight, a lot of critics have taken to dismissing him as the lame member of the team, ignoring how he almost single-handedly took down the Helicarrier. It hasn't been helped by his lack of focus in the films, and being almost completely absent from Phase 2 thanks to scheduling dificulties with Jeremy Renner.