Perhaps he's had a falling out with some of the people at the camp. There could be misunderstandings or conflicts that make him feel unwelcome. For example, if he's wrongly accused of something, he might decide to leave in anger or to clear his name away from the distractions of the camp. Also, he might have received a sign or a vision that leads him to believe he has to leave the camp to fulfill his destiny.
One reason could be that he feels a sense of being out of place. Maybe new campers have come in and he no longer feels as important or needed. He might also leave in search of a greater purpose. In some fanfictions, he could be on a quest to find something that he believes is crucial for the safety of the camp, but he has to leave the camp itself to do it.
In fanfiction, Percy could leave the camp because he's curious about the outside world. He has spent a lot of time at the camp, and he might want to explore what lies beyond. He could be drawn to new adventures, new people, or new powers that he believes exist outside the camp's boundaries. He may also think that by leaving, he can gain more knowledge and experience that will ultimately help the camp in a bigger way.
One possible reason could be a misunderstanding. Maybe there was a miscommunication between Annabeth and Percy about their future plans or how to raise their child. Another reason might be some external threat that Annabeth believes she can face alone to protect her family, so she leaves them behind. It could also be that she has some personal issues to deal with, like an identity crisis, which makes her think she's not fit to be with them for the time being.
Perhaps Percy was kicked out because of a prophecy that the camp elders misinterpreted. They thought that Percy's presence would bring great danger to the camp, so they made the hasty decision to expel him. It could also be that there was some sort of political intrigue within the camp. Some powerful figures at the camp might have seen Percy as a threat to their power and influence, and so they fabricated reasons to get him kicked out.
One reason could be that she's scared. The idea of becoming a mother at that point in her life might be overwhelming, and she might think she can't handle it with Percy. Maybe she wants to protect Percy from the difficulties that come with the pregnancy as well.
One possible reason could be that he feels misunderstood. In the camp, everyone has high expectations of him as a son of Poseidon. If he makes a small mistake, it might be blown out of proportion. For example, if he accidentally causes a small flood during a training session, instead of being understood as still learning control over his powers, he could be harshly criticized. Another reason might be that he's tired of the constant danger at the camp. Every day there seems to be a new threat looming, whether it's a monster attack or some sort of internal camp conflict. He might just want to get away and have some peace for a while.
In some fanfictions, it could be that Percy has new, uncontrollable powers that the camp doesn't understand. They see him as a threat and banish him. For example, if he starts to show signs of a power that was once associated with an enemy of the camp. It's also possible that Percy's loyalty is questioned. If he befriends someone from an opposing group, the camp might think he has turned against them and banish him.
It could be due to a conflict with a particular authority figure at the camp. If he has a major disagreement with a camp leader over how to handle a situation, like dealing with a new enemy. Percy, being headstrong, might decide to run away and handle it his own way instead of following the camp's rules.
In some fanfictions, he might be exiled because of false accusations. Maybe someone frames him for a crime he didn't commit, like stealing a powerful artifact from the camp's vault. The camp leaders, believing the false evidence, have no choice but to exile him.
He might leave because of a personal conflict. Perhaps he has a falling - out with some of the other campers or with the camp leaders. Like if he disagrees with a new rule or policy that the camp has implemented which he thinks is unjust.
One possible reason could be a dark turn in his character. Maybe he's been corrupted by some external force like a powerful and evil magic. Another reason might be a misunderstanding. For example, he believes the camp has betrayed him in some way, so he lashes out in anger.
Another possibility is that in the fanfiction, Percy might feel pressured by the expectations of their relationship. He could be afraid that he can't live up to what everyone expects of him as Annabeth's partner. So, he leaves in an attempt to find himself and figure out who he is without that pressure.