Well, it could be set in an alternate universe where the enmity between Harry and Snape is resolved early. They are forced to work closely together on a dangerous mission. During a tense moment, Harry accidentally falls into Snape's lap. Their eyes meet, and in a rush of adrenaline and new - found respect, Harry gives a quick kiss on Snape's lap, leading to a whole exploration of their unspoken feelings.
Perhaps in a fanfic where magic has an unexpected side - effect. Harry and Snape are both affected by a spell that makes them see each other in a different light. They end up in a situation where Harry trips and lands near Snape's legs. The proximity and the strange new feelings lead Harry to plant a gentle kiss on Snape's lap, and from there, the story could explore how they deal with this new - found physical and emotional closeness.
One possible plot could be a moment of unexpected connection. Maybe Harry is injured during a battle and Snape, in a rare moment of vulnerability, tends to his wounds. As Harry recovers, he impulsively shows his gratitude with a kiss on Snape's lap, which startles both of them and sets off a series of complex emotions and a new dynamic in their relationship.
Harry would probably be extremely embarrassed at first. He might blush furiously and stutter out an apology. Snape, on the other hand, would be shocked and likely try to hide his own confusion. He might initially be angry, thinking it was some sort of prank, but then as he realizes Harry's genuine emotion, he would be at a loss for words.
One possibility could be that Harry has faced some extreme danger or a great loss in the wizarding world. Maybe he lost a close friend in a battle, and Snape, despite their past differences, offers him a moment of comfort. Snape, in this scenario, might be showing a side of himself that Harry has never seen before, a more empathetic side.
One possible plot could be that there is a magical accident or spell that binds Harry to Snape in this 'pet - like' relationship. Snape is initially reluctant but then starts to see the potential in having Harry under his control. He could use Harry to further his own goals, whether it's to gain power within Hogwarts or to take revenge on his enemies. Meanwhile, Harry has to figure out how to reverse the situation while dealing with the new and uncomfortable dynamic between him and Snape. It would be a very different take on the Harry Potter story, full of twists and turns as Harry tries to regain his freedom and Snape tries to maintain his hold on him.
A possible theme could be forgiveness. Harry might be in a position where he has to forgive Snape for all his past misdeeds. And this rather strange act of Snape bouncing him on his lap could be a catalyst for that forgiveness. Additionally, it could also touch on the theme of growth and change, both for Snape, who is showing a different side of himself, and for Harry, who has to adapt to this new side of Snape.
One possible plot development could be that Dumbledore gives Snape to Harry as a sort of guide in the wizarding world. Snape, with all his knowledge and experience, could show Harry things that Dumbledore couldn't. This might include some dark arts knowledge that could be used against Voldemort. It could also be a test for both of them. For Snape, to see if he can truly be selfless in helping Harry, and for Harry, to learn to trust Snape despite their past differences.
One possible plot could be that Harry accidentally falls onto Snape's lap during a chaotic Potions class. Snape is initially angry but then realizes Harry is hurt. So, instead of scolding him, he helps Harry get better while Harry is still on his lap in a rather awkward but kind moment. This would show a different side of Snape and a new dynamic between them.
Well, it could be a rather strange plot. Maybe it's a situation where Harry is injured and Snape has to hold him in this way for some kind of magical healing process. Harry might be in a weakened state and Snape, despite their past differences, has to take care of him. This could lead to an unexpected moment of understanding between them.
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 plot development could be that Harry initially struggles with having Snape under his 'command' in some way. Since they had a very strained relationship in the original series, Harry might be hesitant to trust Snape even if Dumbledore has given him this new role. He could start by trying to figure out Snape's true intentions.
One possible plot could be that they discover they are brothers early on in Harry's time at Hogwarts. Snape then has to balance his duties as a teacher and his newfound family connection. He tries to protect Harry in secret from the threats within the school, like Malfoy and his cronies.