Maybe Snape has been wrongly accused of something, and Harry, being the hero - like figure, is trying to protect Snape from an angry mob. But Snape doesn't know this and his usual intimidating demeanor makes Harry run instead of explaining. It could also be that Harry has a premonition of danger related to Snape, like in a magically influenced situation where Snape is unknowingly a threat to Harry, so he runs.
One possible reason could be that Harry is afraid of Snape's intimidating presence. Snape has always been strict and rather unfriendly towards Harry in the original series, so in the fanfiction, this fear might be amplified, causing Harry to run. Another reason could be that Harry has stumbled upon something he shouldn't have seen that involves Snape, and he's running to avoid getting caught.
He could run to the Forbidden Forest. It's a place full of mystery and danger, but also a good hiding place as it's vast and not many people would expect him to be there. He might find some solace among the magical creatures there.
One reason could be fear. Snape has always been rather intimidating to Harry. Snape's cold demeanor and his past interactions with Harry's father might make Harry feel constantly on edge around him. So, running away could be an instinctual response to this fear.
One possible reason could be Snape's lingering resentment towards James Potter, Harry's father. Snape might see Harry as a reminder of the bullying he endured from James during their school days. So, he takes out his anger on Harry.
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 reason could be that Snape has his own complex sense of loyalty. He may feel that Dumbledore's plans for Harry are too dangerous or not in Harry's best interest in some ways. He has a long - standing relationship with Harry's mother, Lily, and perhaps through that connection, he wants to safeguard Harry from any harm that might come from Dumbledore's actions or decisions.
One possible reason could be Snape's sense of duty to protect Hogwarts students in general. Even though he dislikes Harry, he might put aside personal feelings to ensure the safety of a student in a dangerous tournament like the Triwizard Tournament. Another reason could be Dumbledore's influence. Dumbledore trusts Snape and may have asked him to keep an eye on Harry during this event.
Maybe Snape has some sort of premonition or knowledge that Harry is going to be crucial in the fight against dark forces in the future. So, in order to ensure that future, he steps in to save Harry from whatever danger his parents' situation might put him in. He might also be trying to atone for past mistakes or grudges he held against Harry's father by protecting his son.
One possible reason could be Snape's long - standing grudge against James Potter, Harry's father. Since Snape had a difficult past with James, he might project that onto Harry. Another reason could be that Snape is trying to protect Harry in a very strange way. He may think that by rejecting Harry, he is keeping him at a distance from the dangerous wizarding world politics and plots that Snape is often involved in.
Snape might do it out of his love for Lily. Since Harry is Lily's son, he feels a sense of obligation to protect him.