One possibility is that Minato takes Naruto away to a secret training ground. There, he could start to teach Naruto advanced jutsu that he never got to pass on before. This would be a way for Minato to make up for lost time and help Naruto become even stronger.
It might develop with Naruto being initially confused and a bit angry at being taken away. But as Minato starts to show him new techniques or tell him stories about his past, Naruto would gradually come to understand. For example, Minato could show Naruto how to better control his chakra, which would be a major development in the fanfiction. And as they spend more time together, their father - son relationship would strengthen, which would be a central part of the story's development.
One possible reason could be fear. Minato is a very powerful shinobi, and Naruto might be scared of him for some reason. Maybe he has heard stories about Minato that made him afraid.
Well, one possibility could be that Kushina takes Naruto away to protect him from the dangerous world of shinobi. Maybe she wants to raise him in a peaceful place away from the constant threats and battles in the Hidden Leaf Village.
There might also be other characters involved. For example, Minato might make an appearance in flashbacks. Since he was Kushina's husband and Naruto's father, his presence could be important in terms of the backstory. Also, characters from the Hidden Leaf Village might be mentioned or even try to find Naruto and Kushina, like Iruka Sensei who has a special bond with Naruto.
Naruto could end up getting lost, both literally and metaphorically. He might lose his way in the shinobi world, not having Minato's guidance. He could make wrong decisions about his training, alliances, etc.
One possible consequence could be that Naruto grows up away from the influence of Konoha's shinobi culture. He might develop a different set of values and skills. For example, without the guidance of Iruka and Kakashi, his understanding of ninjutsu and teamwork could be very different. Also, his relationships with characters like Sasuke and Sakura would be greatly affected. Maybe he wouldn't have the same rivalry with Sasuke, and his bond with Sakura might not even form.
In fanfiction, Naruto and Minato often represent a connection between generations. Minato was a great ninja in his time, and Naruto is following in his footsteps. Fanfictions can explore how Naruto lives up to his father's reputation.
I'm not sure if there is such a specific fanfiction. Fanfiction can cover a wide variety of themes and scenarios, some of which might be rather strange or controversial. However, Naruto fanfiction often explores different relationships and alternate storylines, but this particular one seems a bit unorthodox and I haven't personally come across it.
One popular theme is Minato getting to know Naruto in the afterlife. Since Minato died when Naruto was a baby, fanfictions often explore their relationship in a spiritual realm.