One reason could be the loss of his father. In the Naruto universe, Kakashi had a complex relationship with his father's death. If a fanfiction delves deeper into his younger years, this could be a very emotional trigger for him to cry.
Perhaps he might cry over the loss of his comrades. If the fanfiction creates a scenario where he witnesses the death of his friends at a young age, like what he later experienced with Obito and Rin, it would be a very likely reason for him to break down in tears. It shows his vulnerability and the emotional toll that war and loss can take on him.
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 reason could be his regret over his past actions. For example, when he distanced himself from his family to pursue his career at the Ministry. He might realize later how wrong he was and cry out of remorse.
Guilt. For example, if he fails to save someone he cares about. He'll feel responsible and that could lead to him crying.
The threat to his friends. James is very loyal to his friends, like Sirius, Remus, and Peter. If they were in danger or if something bad happened to them, it would be understandable for him to cry. For example, if one of them got seriously injured or was in a life - threatening situation, his emotions would be on full display.
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.
One reason might be that he doesn't like how fanfiction can change the relationships between characters. For example, in the original, his relationship with Naruto is based on training and mentorship, but fanfiction could make it something completely different.
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.