The Mouse Cursor Suddenly Has a Mind of Its Own

Random movement and clicks can come from the mouse, a laptop touchpad or a touchscreen registering touches that are not happening. The quickest way to narrow it down is to disconnect or disable one input device at a time.

1. Disconnect the mouse, including its USB receiver, or switch it off if it uses Bluetooth. If the movement stops, clean the sensor, try a plain mouse pad, replace or charge the battery, and test another mouse.
2. If it continues on a laptop, disable the touchpad temporarily. Connect a working external mouse first, then open Settings > Bluetooth & devices > Touchpad on Windows 11, or Settings > Devices > Touchpad on Windows 10.
3. If the computer has a touchscreen, check for ghost touches. With an external mouse available, open Device Manager > Human Interface Devices, right-click HID-compliant touch screen, and select Disable device. If the unwanted clicks stop, the touchscreen is the likely source. Re-enable it after testing.
4. For a laptop, briefly test on battery power. If it happens only while charging, test with a known-good, manufacturer-approved charger. Charger interference can affect touch input.

If the cursor still moves after the physical input devices have been isolated, close any remote-access or mouse-automation software and run a full scan through Windows Security. Unwanted cursor movement alone does not prove someone has accessed your computer.

It could be a malfunctioning mouse.
Unplug the mouse (or turn it off, if it's wireless).  Is it still happening?
If so, do you have two pointing devices connected (touchpad? Thinkpad trackpoint?)
If not, try cleaning the crud from the bottom of the mouse, or changing the battery, or banging it on your desk, or (better yet!) replacing it.

Perhaps you let a scammer talk you into installng remote access software on your computer.
Do the mouse motions and clicks seem purposeful?
Disconnect the Internet. Is it still happening?
If disconnecting the internet prevents it from happening, then that's probably what's going on, especially if the mouse motions and clicks seem purposeful.
Quickest fix might be a System Restore to before it started happening, if you have restore points enabled. Before doing the system restore, be sure to save any important documents or pictures that you left on the Windows Desktop. System Restore should not change your Documents, Pictures & Downloads folders, but it WILL delete recent files form your Desktop.