Another possibility is a jealous rival. Maybe there's someone who is jealous of Gohan's power or his relationship with certain characters. This rival could do things to annoy Gohan and push him to the edge of his temper.
It could be Vegeta. Maybe Vegeta does something really underhanded like stealing Goku's senzu beans or trying to take all the credit for a victory that Goku also contributed to. Goku, being a fair - minded person, would get angry at such behavior.
Maybe he gets mad when his friends are in danger. For example, if Videl is threatened by a powerful villain, Gohan's protective nature kicks in and he gets really mad.
Maybe Chi - chi is constantly comparing Gohan to Goku in the fanfiction. Gohan has his own identity and abilities, and being compared all the time can be frustrating. He could get mad because he wants to be recognized for who he is, not just as Goku's son. Also, if Chi - chi is interfering too much in his training or friendships, that could also lead to his anger.
Well, perhaps in the fanfiction, Goku gets mad when he discovers that someone has been using his name for evil deeds. This would go against his values of justice and protecting the innocent. He might get really angry and go on a search to clear his name and stop the impostor. Or it could be that a new, extremely powerful villain emerges and starts to destroy planets without any remorse. Goku, being the defender of the universe, would be enraged by such senseless destruction and would confront the villain with all his might.
In some fanfictions, Gohan and Goku might have a series of conversations that open up new levels of understanding between them. Gohan could share his thoughts and feelings about his life and future, and Goku would listen and offer his own insights. This kind of communication can really strengthen their bond.
In such fanfictions, Goku might develop by becoming more of a mentor to Gohan. He could teach Gohan new fighting moves and also how to control his emotions better during battles.
In a Gohan x Goku fanfic, their relationship can be delved into by creating scenarios where they have to rely on each other completely. For example, if they are in a situation where Goku's strength and Gohan's intelligence are both required to escape. Their different personalities can also be highlighted. Goku's carefree nature contrasted with Gohan's more studious and responsible side. And of course, flashbacks to Gohan's childhood and how Goku's presence or absence affected him can add depth to their relationship.
One possible interaction is that Bardock could spar with Goku and Gohan. He'd test their limits and see how much they've learned about fighting. Another could be Bardock sharing stories of Planet Vegeta with them, which would be really interesting for Goku and Gohan to hear.
In the manga, Gohan never managed to defeat Goku. Their battles often ended with Goku coming out on top or the fight being interrupted for some reason.