Can I make a new apple ID to use for one specific game?
Making new Apple ID is easy but hard to manage long term. It almost certainly won't help you wi your game restart for several reasons listed below. 

Unless you have a very special game that somehow asks you to log in and doesn't use Game Center to find your Apple ID, your phone expects all games to use the same Apple ID. 

If the developers stored your progress using iCloud, you can simply delete that game from all of your devices and when the last delete happens, it will also delete the saved progress. 

If that doesn't work (and in many cases - especially "free" games, this won't work) you will need to ask game support to reset your progress or ditch that old Apple ID forever and make a new one for all your games.