Snape might use his potions knowledge to brew a remedy for Harry's illness. Maybe he gives Harry a special potion that eases his symptoms and helps him recover faster.
In this fanfiction, it might be that Snape sees Harry in a vulnerable state during winter break. Maybe Harry is being bullied by some Slytherins even during the break at Hogwarts. Snape, despite his usual dislike for Harry, steps in to protect him. This action could start to change Harry's perception of Snape, and they could form an unlikely alliance. As time passes, Snape could start to share some of his own experiences with Harry, which helps Harry grow not just as a wizard but also as a person.
At first, it would probably be very rocky. Snape has long - held grudges against Harry, and Harry doesn't trust Snape much either. But as time goes on, they might find common ground. Maybe they both love a particular type of magic or have a shared goal of protecting Hogwarts from an unseen threat.
There is no evidence in the original 'Harry Potter' series or any official materials to suggest such a thing. This is a very inappropriate and untrue speculation that goes against the nature of the characters created by J.K. Rowling.
Well, one possibility is that Snape could start to see Harry in a new light. Maybe he discovers Harry's hidden talents or interests during the summer. For example, Harry might show an aptitude for potions that Snape had previously overlooked.
One common theme is vulnerability. When Snape is sick, his usual tough exterior cracks, and we see a more human side of him. Another theme is the exploration of his past. Illness might be related to something from his past, like an old potion experiment gone wrong. Also, the theme of redemption can be present. As he's sick, others might show him kindness, and he may start to change his ways or be more open to others.
One common theme is vulnerability. When Snape is sick, he's not his usual intimidating self, and this shows a side of him that is often hidden. Another theme is the discovery of hidden kindness. Some characters who might not seem sympathetic towards Snape in the original stories, might show concern when he's ill. For example, students who were always a bit afraid of him could end up being very caring.
One possible reason could be Snape's long - standing dislike for James Potter, Harry's father. Since Draco is from a pure - blood family like Snape himself, and Snape has a certain affinity for the pure - blood traditions and values, he might choose Draco. Also, Draco was in Slytherin, Snape's own house at Hogwarts, and he might have felt a stronger sense of loyalty to his house and its students.