She might be initially defensive. Hermione is a proud and independent character. So, when Snape finds out about her arm, she could be worried that he'll use it against her in some way, like taking away house points or using it as an excuse to give her extra work.
Maybe Snape uses this discovery as an opportunity to teach Hermione something new. He could be curious about how she got the condition on her arm and offer to show her a spell or potion to heal or manage it. This would also be a chance for him to show a different side of himself, not just the strict professor.
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.
Snape's reaction could be multi - faceted. He might be angry at those who are abusing Draco, but also a bit conflicted because of Draco's family's ties to the Dark side. He could use his position as a teacher to give Draco some extra guidance and support, perhaps in the form of private tutoring sessions where he can also check on Draco's well - being more closely. He may also start to question his own views on the Malfoy family in general.
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.
Snape might initially be shocked. He's not one to show much emotion openly, but internally he could be very disturbed. He might start investigating what happened to Hermione immediately, using his skills in potions and magic to try and find a cure or at least understand the cause.
She might initially be shocked and scared. But then she would start to think rationally and try to find a way to outwit Snape. For example, she could start to gather evidence against Snape's own wrongdoings, if there are any, so that she can turn the tables on him.
He may show a rare moment of kindness. Given his own past traumas and experiences, he might empathize with Hermione on some level. He could offer her some words of wisdom or even some form of magical assistance to help her get better.
He might immediately go into a protective mode. He could use his potions knowledge to quickly make a healing draught for her. After all, Snape is very skilled in potions, and Hermione being hurt would likely trigger his instinct to help.
He might be initially startled. Snape is used to seeing Hermione as a strong - willed and confident student. So seeing her crying could catch him off - guard. But then he might quickly put on his usual cold facade and ask her curtly what's wrong.
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.