Maybe it's related to Naruto's connection with the tailed beasts. Naruto has a unique bond with Kurama, and he endures a great deal because of it. Kakashi, understanding the pain and the power that comes with such a bond, could cry for Naruto as he witnesses Naruto's journey of taming the power within him and using it for good.
One reason could be Naruto's strong will. Naruto has faced so many difficulties but never gives up. Kakashi, seeing Naruto's perseverance, might be moved to tears as he remembers his own past struggles and how Naruto is so different yet so inspiring. He might cry out of pride for Naruto's growth.
One reason could be that the fanfiction changes Kakashi's teaching methods too much. Naruto is used to Kakashi's unique style in the original story, and if it's drastically altered in a way that doesn't seem right to Naruto, he might not like it.
One possible reason could be that Kakashi is preoccupied with other important matters in the village. Maybe he has some top - secret missions or tasks assigned by the Hokage that demand his full attention, leaving him with little time to focus on Naruto.
One reason could be that he sees Naruto as a younger version of himself in some ways. Both have had their share of loneliness and challenges growing up. Kakashi may want to prevent Naruto from going through the same pain he did. Another reason is Naruto's importance to the village. As the jinchuriki of the Nine - Tails, Naruto is a key asset. Kakashi, being a loyal ninja, wants to protect the village's future which is tied to Naruto.
One possible reason could be that Naruto might feel Kakashi is too strict with his training. Maybe Kakashi's methods are too harsh for Naruto, making him frustrated and gradually developing dislike.
One possible reason could be Kakashi's reputation as a strict and powerful teacher. Naruto might be in awe of his skills and fear making mistakes in front of him.
One possible reason could be that Naruto might be trying to prove his independence. He has grown and wants to show that he can handle things on his own without always relying on Kakashi's guidance.
Maybe it's related to his relationship with Cedric. In the Goblet of Fire, Cedric is a competitor but also a friend in a way. His death is a huge shock in the original story. In fanfiction, it could be that events leading up to Cedric's death or just the general unfairness of the situation makes Harry cry. For example, if there are false accusations against Cedric or if Harry feels he could have done more to prevent the tragedy.
Perhaps the fanfiction alters the relationship between Kakashi and Naruto in a way that Kakashi doesn't like. Maybe it shows Kakashi as being overly harsh or unfair to Naruto in a situation where in the original series he wouldn't be. It could also be that the fanfiction contains plot elements that Kakashi, as a character, would find illogical or against the spirit of the Naruto world.
Perhaps Kakashi feels that Naruto is becoming too reliant on him and not developing his own skills independently. So, in order to force Naruto to become stronger on his own, he decides to abandon him. This is a tough - love approach. Naruto has great potential, but he needs to learn to stand on his own two feet, and Kakashi may think this is the best way to make that happen.
Maybe in a fanfiction where Kakashi is mind - controlled. It could lead him to harm Naruto without his normal self - control.