One of my favorite games is Papers Please, where the premise is pretty much people come through the line at an immigration checkpoint and give you papers, and you have to see if they have all the valid documents. My game idea is similar in the primary mechanic, but more abstract when it comes to the setting.

Your job is to patrol the school for kids that aren't supposed to be in the hall. On your first day, you're only chosen to monitor one hallway. As the game progresses, administration gives you more of the school to monitor, but since you have more work you can either a) be allowed to miss more kids, as you can't be everywhere, or b) set up other kids as posts to help you monitor the school. The game takes place over a set amount of time (30 days/ levels, maybe more)

Once you find a kid in the hall, you interrogate them, they give you a hall pass (each classroom has a different hall pass, and you have to see if its valid or not, this is the primary inspiration from Papers Please). If they have a valid pass, great. If not, you can write them up, or they may run, and you have to chase them before they go inside the bathroom, their classroom, etc. Chasing them down could lead to not interrogating other kids in the hall. Some kids may give you a sob story, or have a "valid" excuse as to why they don't need a pass. Whether or not you believe them is up to you. However, if you miss a kid, or let one go that shouldn't be let go, a teacher may come out of a classroom and tell you that they never get their permission, getting you in trouble. So, you have to predict certain teachers' behavior as you progress (some may not care, some are very very strict).

There can also be an overarching story in the game. Maybe there is a clique in the school that tries to abuse your powers, either through threats or bribes to make sure they don't get tickets. Whether or not you let them operate is your choice.

Tell me what you think, typing it all out makes it sound a lot like Papers Please but maybe that's just because it's inspired off of it. Hopefully I added something more that isn't just a reskin of that game.