Kakashi is known for his laid - back yet highly skilled nature. His influence on Naruto could be in terms of teaching him to be more strategic in battles. Rin, being kind - hearted and having a strong sense of care, would give Naruto the emotional support he lacks. So Naruto might become more balanced in his approach to life and ninja - hood.
In this fanfiction, perhaps the story could focus on how Kakashi and Rin's adoption of Naruto changes the relationships in the village. Naruto might not be as ostracized as he was in the original story. Also, they could help Naruto deal with the Nine - Tails inside him in a more gentle and effective way compared to what happened in the canon story.
In a fanfiction of Kakashi and Rin's love story, it could start with their time in the ninja academy. Rin was always drawn to Kakashi's natural talent and mysterious aura. Kakashi, on the other hand, was initially oblivious to Rin's feelings. One day, Rin got injured during a training session. Kakashi carried her all the way to the medical tent. That was the first time he realized how much she meant to him. As they grew older, their bond deepened, and they found themselves falling in love while on various missions.
However, in the realm of fanfiction, anything can happen. A writer could create an elaborate backstory where Naruto somehow ends up in the Blue Exorcist world and becomes Rin's father. Maybe there could be some sort of interdimensional travel or a really wild magic spell that links the two universes in a way that makes this relationship possible. But it would require a lot of creativity to make it believable within the context of the combined universes.
In fanfiction, anything is possible. However, in the original Naruto series, Kakashi is not Naruto's father. But in the realm of fan - created stories, an author could come up with an interesting plot where Kakashi is Naruto's biological father, perhaps by creating some elaborate backstories and twists to make it seem plausible.
In a fanfiction, Kakashi might reveal to Naruto that he knew his father very well. He could share stories about Minato, which would make Naruto feel closer to his dad. Kakashi could also give Naruto a special gift that once belonged to Minato, symbolizing the connection between them as a father - son - like relationship.
There are many interesting fanfictions on this topic. Some might explore the relationship between Kakashi's father, Sakumo Hatake, and the events in the Naruto world that we know of. It could delve into his past missions, his influence on Kakashi's upbringing even though he passed away early. Maybe in some fanfictions, he has a connection to some of the main plotlines that Naruto is involved in, like the overarching story of the shinobi world and its conflicts.
It would be really strange. Naruto is much younger than Kakashi in the original story. But if it were a fanfiction, maybe it could involve some time - travel or alternate universe concept. Naruto could use his experiences to raise Kakashi in a different way than Sakumo did.
I'm not sure of a specific 'Naruto father of Kakashi' fanfiction. However, you can try searching on popular fanfiction websites like FanFiction.net or Archive of Our Own. They have a vast collection of Naruto - related fanfictions and you might be able to find one with that particular concept there.