The plot could be that Dumbledore assigns Harry to clean Snape's home as a way to build a bond between them. Harry initially resents it but as he cleans, he finds a diary that reveals Snape's unrequited love for Lily. This discovery makes Harry see Snape in a new light and he decides to try and befriend him. He starts to clean with more care, and when Snape returns, he is initially angry but then realizes Harry has a newfound respect for him.
A plot could involve Harry being under the influence of a spell that makes him think Snape's home is in desperate need of cleaning. So he barges in and starts cleaning. Snape is away on a mission at first. When he returns, he finds Harry has completely rearranged his potions lab while cleaning. Snape is about to hex Harry but then realizes that Harry has actually improved the efficiency of his lab in some ways. This leads to a strange alliance between them where Harry helps Snape with his potions research in exchange for Snape teaching him some advanced magic.
In this fanfiction, perhaps the dog that bit Snape was under some dark magic influence. Harry not only cleans the physical bite but also has to figure out the source of the dark magic. He might have to search for clues in Hogwarts, and along the way, he and Snape start to understand each other better.
Well, perhaps Harry discovers Snape's dog has bitten someone. He might initially be reluctant to get involved but his sense of justice makes him step up. He could start by finding out how severe the bite is and then look for a way to clean the wound properly to prevent infection.
Perhaps it's one of the Hogwarts students. The dog might have been provoked by the student's actions, and Snape is too embarrassed or proud to clean the bite himself, so Harry is made to do it.
It could be an experienced fanfic author who has written a lot about the Harry Potter series. They might have decided to write this particular story to add a different kind of interaction between the two characters. Maybe they noticed that there are not many stories about Harry helping Snape in such a specific way.
The plot might start with a magical experiment gone wrong. Harry ends up with a special mark or power that makes him Snape's master. As the story progresses, Harry uses this power to make Snape teach him advanced potions and dark arts defenses. Snape resents it but has no choice. Meanwhile, Voldemort senses this strange new power dynamic and tries to use it to his advantage. This forces Harry and Snape to work together in ways they never thought possible to outwit Voldemort while also dealing with their own difficult relationship.
Maybe in the fanfiction, Voldemort devises a plan where Snape forgets Harry to disrupt the Order of the Phoenix's plans. Without Snape's knowledge of Harry and his role in the prophecy, things start to go awry. Harry has to find a way to make Snape remember, perhaps by going on a quest to find a magical relic that can restore memories. Hermione and Ron would be by his side, of course, providing their usual support and knowledge.
Well, he might discover some hidden secrets while cleaning. Maybe he stumbles upon a long - forgotten spell or a mysterious object that has been lost for ages.
In a 'Harry sold rescued Snape' fanfiction, the plot could start with Harry learning that Snape is held captive by a group of dark wizards. Harry realizes that he needs certain magical supplies or potions to break Snape free. Since he doesn't have the funds to buy these items, he decides to sell a rare spell book that he found in the Room of Requirement. This spell book has some unique spells that could potentially be used against the dark wizards. But once he sells it, he discovers that the buyer has some ulterior motives and is somehow connected to the group holding Snape. Now Harry has to not only get the spell book back but also figure out how to use it to rescue Snape while dealing with the new threats that have emerged.
Well, it's hard to say exactly without reading an actual piece of such fanfiction. But perhaps it could involve Harry being in some sort of difficult situation where he has to force Mary Snape (assuming she's an original character) to do something for the greater good, like revealing a secret that could help in the fight against dark forces. Or maybe it's a misinterpretation situation where Harry thinks Mary Snape is hiding something dangerous and he has to use force to get to the truth.
One possible plot could be that Snape, despite his initial resentment towards James Potter (Harry's father), sees the vulnerability in little Harry. He decides to take him in out of a sense of duty to Lily (Harry's mother). Snape then has to adjust his life to care for a child. He teaches Harry potions at an early age and tries to protect him from the darker elements in the wizarding world. Harry grows up with a different perspective on Snape than in the original story.