This is a *Solution* to a problem I had with the Kensington Trackball Wireless Mouse
I have had almost continuous problems getting Bluetooth on an iMac M1-M3 to recognize my Kensington Trackball Wireless Mouse after initial boot of my iMac. Only after a 5-second long "wiggling" of the Cursor did the Cursor "unfreeze" from the top-left corner of the screen. After this, everything worked great .. but only after this initial "wiggling".
The solution .. when I initially paired this Trackball Mouse a long time ago, I disconnected the Apple installed wireless mouse that came with the iMac ... but I left it active even though it was disconnected.
When I selected "Forget this Device" for the Apple Mouse, the necessity for the above "wiggling" and the inherent 5-second delay **disappeared** .. now, after a fresh boot of my iMac the Kensington TrackBall Cursor is immediately moveable.