Well, Harry leaving for America in fanfiction could lead to him trying to adapt to a new culture. He might struggle with the differences in magic use between the two countries. For example, the way spells are pronounced or the types of potions that are popular. He could also get involved in some sort of mystery that is unique to the American wizarding community.
He might meet American wizards who are very different from his British friends. Maybe some tough - talking Aurors who have their own way of dealing with dark magic.
Perhaps he leaves for personal growth. America has become a place full of painful memories for him, like the loss of friends during battles that happened on American soil. So, he decides to start anew in another place.
Yes, there might be such fanfiction in the vast world of fan - created works. Fanfiction often explores unique and sometimes out - of - the - ordinary scenarios. However, this particular concept seems rather unconventional and might not be as common as more traditional fanfiction themes within the Harry Potter fandom.
As he leaves, he could face challenges on his journey. Maybe some villains try to take advantage of his vulnerable state during the move. He has to use his wits and some basic tech he has with him to fend them off.
One possible fanfiction could be about Harry Potter visiting America and encountering a new form of dark magic related to death. Maybe he goes to a haunted place in America, like an old plantation, and has to deal with a death - based curse.
Harry might be initially in shock. He can't believe Ginny would leave him and their child. He could become withdrawn, not really knowing how to deal with this new situation.
Yes, there are some out there. You can try searching on fanfiction - specific websites like Archive of Our Own. Just be sure to read the tags and summaries carefully to find one that suits your taste.
One possibility is that Harry discovers some dark secret about Severus that makes him feel he can no longer stay in the same place as him. For example, if he finds out Severus had a hand in some past event that endangered his friends. So he decides to leave to protect himself and those he cares about.