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.
Another possibility is internal injuries from a battle. Fanfiction often has intense battles that are not part of the original story. Harry could have been hit by a powerful spell during a fight that caused internal damage, which then makes him cough up blood.
Coughing up blood referred to coughing up blood from certain parts of the respiratory tract in the lungs. Common causes include chronic asthma, dilated bronchi, respiratory diseases, and cardiovascular diseases. The symptoms of hemoptysis may be bright red or pink blood, which may contain mucus. The amount of hemoptysis is usually small, but if there is a large amount or frequent hemoptysis, you should seek medical attention to determine the cause and receive appropriate treatment.
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, it could be a result of a dark spell cast on him by a Death Eater. Maybe they found a new and insidious way to harm him without leaving an obvious trace.
It could be a new dark wizard. There are always new threats emerging in the fanfiction world. This new dark wizard might have a vendetta against Harry and uses a special spell or potion to make him cough up blood.
One interesting fanfiction could be 'The Darkening Plague' where Harry's coughing blood is a sign of a dark curse spreading in Hogwarts. It has a great plot that weaves mystery and magic.
It could lead to a discovery of a hidden plot against Harry. His coughing up blood might be a sign that there is a much bigger conspiracy at play, perhaps involving someone close to him who wants to weaken him for some ulterior motives.
It could be due to a dark curse or spell. Maybe he was attacked by a powerful dark wizard using a forbidden spell that affects his internal organs and causes bleeding.