Neither my wireless Magic Mouse, nor my battery operated mouse will work with my Mac Mini, despite many attempts, including unplugging, etc. The mouse curser appears on the screen but doesn't click. The Magic Mouse isn't recognized, however they both continue to work fine with my MacBook Air
I think you are referring to your mouse. Your cursor is the little black arrow on the screen. The mouse is the physical hardware that you hold in your hand.
Terminology aside, there are several points of clarification that we need from you, in order to help you better.
If you are referring to Apple's Magic Mouse, then there is nothing to "unplug" to reset. (other than the removal of the batteries.) The first thing you should check is that Bluetooth is turned on under System Preferences. You can search
Bluetooth Preference in spotlight, and it should be the first hit.
Note that a bluetooth mouse can only be paired to one computer at a time. If you paired it to your MacBook Air, it will no longer be associated with your Mini, until re-paired with your Mini. At that point, it will no longer work on your MacBook Air.