There's a chance it could be an original character created for the fanfic. A dark wizard who has a vendetta against Snape for some past wrong (maybe something that happened in the fanfic's unique backstory) and is determined to end Snape's life.
It could be Bellatrix Lestrange. She was always a fanatic follower of Voldemort and hated those who she thought were against the Dark Lord. Since Snape was a double - agent, she might have discovered his true loyalties and decided to kill him in the fanfic.
Well, it could be that Snape is wrongly accused of a great crime in this fanfic, and in the heat of the moment, someone attempts to kill him. Maybe it's a misdirected act of revenge by someone who thinks Snape is behind a series of dark events at Hogwarts. But of course, the truth might be far more complex and intertwined with the overall politics and secrets of the wizarding world.
Perhaps in the fanfic, Snape discovers a powerful and dangerous magic that he wants to use for what he believes is the greater good, but Harry sees it as a threat. Harry has always been wary of Snape, and this new discovery could be the tipping point. Also, if Snape were to make a deal with a dark force in order to gain knowledge to defeat Voldemort but Harry misinterprets his intentions, it could lead to Harry killing Snape.
One possible motive could be a misunderstanding. Maybe Snape is framed for something really bad, like a dark magic plot against Hogwarts. Harry, in his haste to protect the school and his friends, believes Snape is guilty and takes drastic action. Another reason could be if Snape accidentally endangers someone Harry loves dearly, like Ginny or Ron, and Harry's emotions get the better of him in the heat of the moment.
It could be in a dark alternate universe fanfic where Harry is driven to extreme actions due to unforeseen circumstances.
One possible motive could be that Snape is still deeply influenced by his loyalty to Voldemort in this fanfiction. Maybe Voldemort has some hold over him that forces him to do the unthinkable. Another reason could be a misunderstanding. Snape might believe that Harry has turned to the dark side or is about to, and he decides to take preemptive action to stop what he thinks will be a great evil.
Hermione would likely be there. She's Harry's best friend and is always ready to support him. So, when Harry talks back to Snape, she might be there in the background, giving Harry encouraging looks or even stepping in to add her own two cents if things get really out of hand. Ron could also be present. He's not one to shy away from a confrontation, especially when it involves his friends. He might be the one to start a bit of a ruckus among the other students as Harry and Snape have their exchange.
It could be Dumbledore from beyond the grave. He might find a way to reach out and summon Harry, believing that only Harry has the potential to truly defeat Voldemort in this new, dire situation.
It could be Jon Snow. He had to grow up a lot in the series and leave behind his more boyish ways. In a fanfic with this title, it would make sense for him to be the main character as he constantly had to make tough decisions that forced him to shed his innocence.
Obviously, Harry Potter would be one of the main characters. He's the one who is likely to be under Snape's desk for some reason. And of course, Snape himself. His desk is the central location, so he has to be involved. Also, Hermione might be there. She often helps Harry out, and if Harry gets into trouble under Snape's desk, she could be the one to figure out a solution.
Harry Potter could visit Snape in the hospital. Despite their past differences, Harry might feel a sense of obligation or curiosity. He could be there to finally have an honest conversation with Snape and perhaps gain a better understanding of his actions throughout the years.