Harry might start by making a list of all the wrongs Snape has done to him over the years. This would include things like unfair grading, public humiliation, and general nastiness. Then, he could look for ways to expose these actions to the rest of the wizarding world. Maybe he would use the wizarding newspapers or even start his own blog (if there were such a thing in the wizarding world) to share his experiences. He could also try to find allies among the students and faculty who have also had problems with Snape.
Harry could take a more psychological approach. He could constantly question Snape's loyalty in public, bringing up his past as a Death Eater. He could make Snape feel like he's always being watched and judged. Harry could also try to turn Snape's own house against him. By highlighting Snape's unfair treatment of Slytherin students, he could create unrest within the Slytherin common room. This would not only cause problems for Snape within the school but also damage his reputation as a Slytherin head of house.
He could use magic in creative ways. For instance, he could charm Snape's office so that every time Snape enters, a loud and annoying noise plays. Or he could use a spell to make Snape's quill write embarrassing things instead of what he intends. Harry could also try to get into Snape's personal memories. If he manages to do that, he could find some truly compromising information that he could use against Snape. However, this would be a very difficult and unethical spell to use, but in the world of fanfiction, anything is possible.
In some versions, Snape might use his knowledge of the Dark Arts to expose the wrongdoings of those who have wronged Harry. For example, if there are Death Eaters or bullies in Hogwarts who target Harry, Snape could reveal their secret plans or actions that are against the school rules or the wizarding law.
In some fanfictions, Harry might expose Snape's past wrongdoings to the whole school. For example, if Snape had been unfair in grading or showing favoritism to certain students, Harry could find evidence and make it public. This would damage Snape's reputation among the students and teachers.
In some fanfictions, Snape might share some of his knowledge about the wizarding world's secrets with Harry. Maybe he tells Harry about certain spells or magical artifacts that could help Harry in his fight against Voldemort. This act of sharing knowledge is a way of being nice as it gives Harry an advantage in his battles.
In some fanfictions, Snape might use his advanced knowledge of potions to create a special concoction to heal Harry. He could draw on his skills in Herbology as well to find rare ingredients for the potion.
Harry could also outwit Snape by using his quick thinking and ability to improvise. Suppose there is a situation where Snape is trying to catch Harry doing something wrong. Harry might use the chaos around him, like a commotion caused by other students, to his advantage. He could blend in with the situation, making it seem like he's innocent while actually achieving his own goals, like getting access to a restricted area for a noble reason that Snape doesn't initially understand.
As the story progresses, they might start to work together more closely. Snape could be teaching Harry some advanced magic that no one else knows, and through this process, they build a bond. Harry begins to understand Snape's past better and realizes that they have more in common than he thought, like their love for Lily Potter in different ways.
In this kind of fanfiction, Harry often becomes more proficient in potions and other areas that Snape is good at. His relationship with Snape also makes him more empathetic towards Snape's past and the Slytherin house in general. He may not be as quick to judge people based on their house. Also, he might develop a more complex view of the wizarding world's politics, as Snape is deeply involved in some of the darker aspects of it.
In some fanfictions, Snape might rescue Harry Potter by using his knowledge of the Dark Arts to counter a dark spell aimed at Harry. For example, if Voldemort or his Death Eaters were about to cast a fatal curse on Harry, Snape could step in and deflect it with a powerful counter - curse. He is a very skilled wizard, after all.
In some fanfictions, yes. Writers often explore the complex relationship between Harry and Snape. They might create scenarios where Harry discovers Snape's true loyalties and motives earlier than in the original books, leading to a new level of trust.
The flaw in the plan can completely change the direction of the plot. If there's a flaw in a plan to rescue a captured character, it can lead to the character being held longer, which might make them learn some dark secrets of the enemy, thus changing the power dynamic when they are finally rescued. It can also create more conflict. For example, if the flaw in a plan to win the Quidditch Cup is discovered, it can lead to in - team fighting and distrust among the players, adding a whole new layer of drama to the story.