In some fanfictions, Clint could be a mentor - like figure to Peter. He might teach Peter skills related to combat or survival, given Clint's own set of abilities.
In this fanfiction concept, Clint might also have a protective relationship with Peter. Considering Peter's young age and inexperience in some versions, Clint could be there to keep him out of unnecessary danger. He may use his knowledge of the superhero community and its threats to shield Peter from potential harm.
One way Clint could be a father figure to Peter in fanfiction is through shared experiences. They both have had to deal with being part of a team and having a sense of responsibility. Clint could share stories of his early days in S.H.I.E.L.D. with Peter, which would help Peter understand the history and politics of the superhero world. Also, Clint could be there for Peter during tough times, like when Peter is doubting himself or facing difficult choices. He can offer advice based on his own life, like how to balance normal life with the life of a hero.
Clint might become a father figure to Peter in fanfiction when he notices Peter's potential and takes him under his wing. Maybe Peter is struggling with some aspect of being a hero, like controlling his powers or dealing with the media, and Clint steps in to help.
One possible storyline could be Clint training Peter in archery skills. Since Clint is known for his archery, he could pass on his expertise to Peter, and they could go on adventures together where Peter uses these new skills.
One great 'Peter x Clint Fanfiction' is 'The Unexpected Bond'. It has a really engaging plot where Peter and Clint start off as just teammates but gradually develop a deep friendship that could potentially turn into something more. The way the author describes their interactions, from their training sessions together to the quiet moments of understanding, is really well - done.
One challenge could be the difference in their lifestyles. Peter is used to the city life in Queens, while Clint is more of a loner who has been on various missions around the world. Their daily routines and environments are so different that it might be hard for them to find common ground at first.
One common theme is mentorship. Clint often acts as a mentor to Peter, guiding him in the superhero world.
In some versions, it could be through a chance encounter with old SHIELD files. Maybe he stumbles upon some classified documents while helping out at the new Avengers facility. The files might have been misfiled or left unguarded for a moment, and his inquisitive nature leads him to dig deeper.
It could be about Peter Parker getting into some sort of situation where he's freezing, like maybe he's lost in a cold place or his powers are malfunctioning and Clint comes across him and has to help. Maybe Clint uses his skills to keep Peter warm until they can get to safety.
The plot could involve Clint taking pity on Peter and trying to help him get back on his feet. Maybe Clint offers Peter a place to stay, food, and helps him figure out what went wrong that led to his homelessness.
Archive of Our Own (AO3) is a great place to start. It has a large collection of 'Peter x Clint Fanfiction'. You can use the search function and filter by tags like 'Peter Parker', 'Clint Barton' and 'relationship' to find exactly what you're looking for.