It could be that Harry, through his kind heart, comes to understand Severus's complex past and motives. He realizes that Severus is not all bad and that there is a deeper connection between them. This new - found understanding and emotional bond drives Harry to save Severus from harm.
In this fan fiction, Harry might save Severus through some form of unique bond. It could be an emotional bond where Harry realizes Severus's hidden goodness or a magical bond that gives him the power to rescue Severus from a dangerous situation, like a dark spell or a life - threatening attack from Voldemort's side.
Maybe Harry has some new information about a greater threat that would also endanger Voldemort. So, he asks Voldemort to save him so that they can join forces against this new threat. Like a new dark entity from another dimension that is coming to destroy the wizarding world, and Harry figures Voldemort wouldn't want to lose his own power base.
It could be the pressure of being 'The Chosen One'. Harry has so much expected of him, and the constant fear of failing in his fight against Voldemort might be overwhelming.
One reason could be redemption. Harry realizes that Snape has had a very difficult life and has been on the right side all along, despite his harsh exterior. Harry, being the kind - hearted person he is, would want to save Snape to give him a chance at a better life after all the sacrifices he has made for the wizarding world. For example, Snape's love for Harry's mother Lily was so strong that he dedicated his life to protecting Harry, so Harry might feel a sense of indebtedness.
One possible reason could be Severus' long - standing grudge against James Potter, Harry's father. Since Severus had a very difficult relationship with James, his hatred might be so deep that it clouds his judgment when it comes to Harry, who is a reminder of James. Another reason could be Severus' loyalty to Voldemort in the past. He might fear that accepting Harry as his son could put him in a difficult position with the Death Eaters or those who still hold onto the old Dark Arts ideology.
One reason could be that he realizes Harry has a lot of potential and wants to nurture it. He sees that Harry is in a difficult position, being the target of Voldemort, and decides to help him despite their past differences.
One possible reason could be that Snape still has an underlying sense of loyalty to Dumbledore. Since Dumbledore cares about Harry, Snape might step in to save him from Umbridge's cruelty. He doesn't want to see Hogwarts turned into a place of unjust persecution under Umbridge's regime, and Harry is a symbol of the resistance against that.
One reason could be his sense of duty. Despite his dislike for Harry on the surface, he might still feel responsible for the safety of all students at Hogwarts, including Harry. So when he sees Harry in danger from Dementors, he steps in to save him.
One possible reason could be Dumbledore's manipulation. Dumbledore might see it as a way to keep Harry safe under Severus' watchful eye, knowing Severus' skills in magic and his loyalty (in his own complex way). Another reason could be a magical binding or an old wizarding law that, due to certain circumstances, forces Severus into this situation. For example, if Harry's magical protection is somehow linked to Severus in a way that adoption is the only solution to maintain that protection.
Isolation. Snape is a lonely character. He doesn't have many friends, and he is constantly in a difficult position between the Death Eaters and the Order of the Phoenix. His feeling of being alone and not being fully understood by anyone can lead to crying in fanfiction.