my magic mouse is sluggish and hangs sometime when I try to move it I have 2 Magic Mice, Mouses.whatever.. One runs on 2 AA batteries and the other one is rechargeable. I use the 2-AA one when I'm recharging the other.
Recently the rechargeable mouse has started functioning strangely. The movement is jumpy and it seems to stick at times. It's hard to do fine movements. I cleaned the visible sensor used for tracking but it had no effect. If I switch to my old 2 battery mouse it works fine. What can cause one Bluetooth mouse to work and another one not? They both worked fine about a week ago when I had to charge one.

Do the System Preferences settings apply for any mouse being used or are they mouse-specific.
There is a known problem with Bluetooth being interfered with by WiFi and especially by USB3 devices. This seems to be a particular problem with the 2018 Mac Minis. See this discussion. 

What Mac model?