He could also be a bit intrigued. Harry running away might mean there is something bigger going on that Snape doesn't know about. So, he might start investigating more thoroughly, using his skills as a Potions master and his knowledge of Hogwarts' secrets. He may even use his spy skills that he developed when he was working for Dumbledore.
Snape's reaction would be complex. On one level, he would be annoyed and perhaps feel that Harry is being a typical, disobedient student. However, he also has a sense of duty. He would realize that Harry is an important part of the wizarding world's future, especially in the fight against Voldemort. So, he might try to find Harry not just to bring him back to Hogwarts, but also to protect him from whatever danger made him run away in the first place. He would be torn between his dislike for Harry and his understanding of Harry's significance.
Snape would likely be angry at first. Harry has always been a source of trouble in his eyes, and running away would just add to that perception. He might mutter some sarcastic remarks under his breath about Potter's irresponsibility.
Snape might suspect that there is some dark magic at play. Given his knowledge and expertise in the Dark Arts, he would immediately look into any signs of foul play. He could also be conflicted because on one hand, Harry is a student he has had many issues with, but on the other hand, he is also an important figure in the fight against Voldemort. So he would have to decide whether to report Harry's disappearance or try to handle it privately.
He could be secretly worried. Snape might use his knowledge of potions and dark magic to try and find a cure or at least a way to stop the bleeding. He may also be conflicted, as his past relationship with Harry and his loyalty to different sides in the wizarding world play a role in how much he wants to help Harry.
He would probably be in shock. Snape has spent so much time around Harry in one way or another throughout his years at Hogwarts. His immediate reaction could be disbelief. He might stand there frozen for a moment, unable to process what he's seeing. Then, the full weight of the situation would hit him, and he could break down in a rare display of raw emotion.
In this kind of fanfiction scenario, Snape might initially be conflicted. On one hand, he has his long - standing dislike for James Potter, Harry's father. But on the other, his sense of duty as a Hogwarts professor could kick in. He might reluctantly start to heal Harry, all the while muttering under his breath about how much trouble Harry always seems to get into, just like his father. Then, as he heals Harry, he might start to see some of Lily, Harry's mother, in him and soften a bit.
Well, Snape has always had a complex relationship with Harry. When he finds Harry cutting himself in this fanfiction, he might be angry at first, thinking it's a sign of weakness. But then, as he realizes the depth of Harry's pain, he could soften. He might start to see Harry more as a troubled child rather than just James Potter's son. Snape could use his knowledge of potions to create something to heal the physical wounds and also try to offer some form of emotional support, albeit in his own gruff way.
Snape's reaction could be quite mixed. On one hand, he might be angry at Dumbledore for not doing more if he knew about the abuse. He may also feel a strange sort of empathy towards Harry. After all, Snape himself had a difficult childhood. So, he could start to be a bit more lenient with Harry in his classes and try to subtly offer support.
He is often very suspicious. He might immediately think it's some kind of trick by the Marauders or their descendants. He'll likely try to figure out how to use it to his advantage against those he dislikes.
Snape would probably be very conflicted. On one hand, he has a long - standing grudge against James Potter, Harry's father. But on the other hand, he loved Harry's mother, Lily. So, he might be cold at first, but then his sense of protecting Lily's son could kick in. He would likely be very strict with Harry, as he is with all his students, but also secretly keep an eye on him to make sure he's safe.
In the fanfic, Snape's reaction could be multi - faceted. He might start by investigating the abuse more deeply. He could use his skills as a Potions master and a former Death Eater to find out who is behind the abuse. And then, he might take steps to ensure Harry's safety. This could involve enlisting the help of other teachers or even trying to get Dumbledore more involved. He may also start to teach Harry some extra skills or give him potions to help him deal with the emotional and physical trauma of the abuse.
Snape might be suspicious at first. He might think it's some sort of trick or part of a bigger plot. However, as he observes Hermione more closely and realizes the genuineness of her situation, his attitude could gradually change to one of reluctant assistance. He might even offer some words of wisdom, albeit in his typical sarcastic way, to help Hermione regain her strength.