In the fanfiction, Naruto could leave Hozuki Castle because he feels out of place there. Despite his friendly nature, the culture or the people in the castle might be too different from what he's used to. He might long for the familiar environment of Konoha or his friends, and this homesickness drives him to leave, even if there were unfinished tasks at the castle. His departure could then lead to a journey of self - discovery as he tries to figure out where he truly belongs in the ninja world.
In the 'Naruto leaves after Hozuki Castle' fanfiction, he could be leaving because he's completed a mission there. He might be leaving behind some sort of mystery or unfinished business. Perhaps there was a powerful item or secret at Hozuki Castle that Naruto either uncovered or was unable to fully understand, and his leaving is to seek more knowledge or help regarding it.
Perhaps they are forced to leave due to false accusations or being framed for something they didn't do. This could create a situation where they feel they have no choice but to leave Konoha in order to clear their names and also protect the village from the real threats that might be lurking in the shadows.
One possible reason could be a mission. Maybe the Hidden Cloud Village (Kumo) requests Naruto's help specifically for a very important and dangerous mission that can only be accomplished by him. So, he has to leave Konoha for Kumo.
One reason could be that he feels ostracized. Despite his efforts to protect Konoha, some villagers still look at him with suspicion because of the Nine - Tails inside him. He might just get tired of constantly having to prove himself and decide to leave in search of a place where he can be accepted for who he is.
One common theme is mystery. The Hozuki Castle is often depicted as a place full of secrets waiting to be uncovered. Another theme is power. There might be powerful jutsus or artifacts hidden within the castle that the characters seek.
One possible reason could be to seek stronger opponents outside of Konoha. Naruto might feel that he has outgrown the training opportunities in the village and wants to face greater challenges.
Perhaps Naruto discovers a dark secret about Hozuki Castle that the rulers don't want to be known. In an attempt to expose it, they turn the tables on him and make it seem like he's the one betraying the castle. This could lead to Naruto having to fight not only the castle's forces but also his former friends who have been misled.
He might face some really tough opponents. In the Naruto world, Hozuki Castle is known for being a dangerous place. Naruto would probably use his signature jutsu like Rasengan to defend himself. Maybe he'd also try to make allies among the inmates there.
Maybe Naruto realizes that his feelings for Sakura are not as strong as he once thought. In the fanfiction, he could have met someone else or had an epiphany about his own emotions. He might leave Sakura to find his true self and explore different relationships.
Maybe Naruto discovers that the village he has been trying so hard to protect has been using him all along. He might find out some dark secrets about the Hidden Leaf, like they were involved in something immoral and were just using his power, which would shatter his idealistic view of the village and leave him heartbroken.