In the comics, Kate Bishop isn't Hawkeye's daughter. They are more like colleagues or partners in the fight against crime, with Hawkeye guiding and inspiring her.
She could have enhanced physical abilities like Black Widow, such as agility and strength. And perhaps some form of Hawkeye's amazing accuracy, not just with a bow but maybe with other weapons or even in her movements.
In a fanfiction, their daughter could have a unique personality. She might be more of an introvert, using her skills in a more stealthy way. She could also have a complex relationship with the other Avengers. Since her parents have such different backgrounds, she could struggle with finding her own identity within the superhero community. Maybe she doesn't want to be just known as Black Widow and Hawkeye's daughter, but wants to make a name for herself with her own unique abilities.
On the other hand, if some of the fanfiction was a bit too outlandish or inappropriate, it could make things a bit awkward. Hawkeye might feel embarrassed, and the other Avengers might not know how to react. But they'd probably try to smooth things over and remind Hawkeye that it's just the creativity of the fans and not a true representation. They'd still support him as a teammate.
There are a few pirate novels with Hawkeye as the main character that I can recommend. Among them were One Piece: Demon Farm, One Piece: Supreme Swordsmanship, and New Pirate: Hawkeye Is My Defeat. These novels were all light novels or doujinshi novels with interesting plots. However, due to the limited search results provided, there might be other pirate novels with Hawkeye as the main character that were not mentioned.