A 'Harry Primary School Fanfic' could involve Harry using his magic to solve everyday problems at school, like when the school's football gets stuck in a tree too high for anyone to reach. He uses a levitation spell secretly to get it down and becomes a bit of a hero among his classmates.
Maybe in the fanfic, Harry attends a primary school that has a secret society of students who have a bit of latent magic in them. Harry is the only one who really knows how to control it well, so he has to teach them in secret, all while trying to fit in and do normal school things like homework and school plays.
One idea could be Harry starting at a new primary school where he discovers a hidden magical garden on the school grounds. It's a place only he can enter because of his special wizarding abilities, and he makes friends with magical creatures there.