One reason could be Harry's impulsiveness. Harry often jumps into dangerous situations without thinking, and Sirius, being his godfather and having a protective nature, would shout at him to make him realize the risks. For instance, if Harry decides to go after a Death Eater alone.
Sirius could help Harry cope by sharing his own experiences of being an outcast in his family. He could tell Harry that just because his parents did this horrible thing, it doesn't mean he's unlovable. Sirius could also introduce Harry to his friends in the wizarding world, like Remus, who could also offer support and friendship.
It's also likely that they go to Romania. Given Sirius' knowledge of the wizarding world, he might know of a safe haven there. In Romania, they could be around dragons and learn about dragon - related magic. Harry would surely be fascinated by the dragons and it could be a place where he can grow stronger in a different kind of magical environment.
It's possible that in the cave, they start to see each other as equals rather than just a mentor - student relationship. They might argue about the best way to deal with the cave's mysteries at first, but through that process, they learn to value each other's perspectives. Also, if one of them gets injured in the cave, the care and concern shown by the other would enhance their relationship.
The relationship could become strained. Sirius might start to view Harry differently, with a bit of wariness. He might not trust Harry as much as before in terms of making good decisions. But it could also lead to a more open relationship where they talk about things they previously avoided, like relationships and feelings.
Well, at first, Sirius would be in a state of shock and panic. But then he'd have to step up. Their relationship would shift from one of equals (in a sense, as Harry was growing up) to a caretaker - cared - for dynamic. Sirius would have to learn about Harry's likes and dislikes as a baby. He'd sing lullabies to him, comfort him when he cries. This new experience would make their bond stronger in a very different way. Harry might not remember all of it when he's turned back, but there would be a subconscious change in how he feels towards Sirius, a deeper sense of security and love that stems from those baby - days when Sirius was his whole world.
The relationship could also take a more complex turn. Other characters might get involved in their conflict. Some might side with Sirius, thinking Harry was in the wrong, while others might try to mediate. This outside influence could either help or hinder the mending of their relationship, depending on how it's handled in the fanfic.
It may explore the emotional bond between them. Sirius could be like a father figure to Harry. He gives Harry advice and guidance, which is different from what Harry gets from others. This special connection makes Harry trust Sirius more than anyone else.
Their relationship would be severely damaged. Harry would likely feel betrayed by Sirius. After all, Sirius is supposed to be one of his protectors. It would be a huge breach of trust. Sirius, on the other hand, would probably be filled with guilt if he realizes his mistake. He would have to work hard to regain Harry's trust, if that's even possible in the context of the fanfic.
Sirius helps Harry become more confident. By sharing his knowledge and experiences, Harry feels more capable. For example, when Sirius teaches Harry a difficult spell, Harry realizes he can master things he thought were out of his reach.