This project shows how to control the mouse using a gamepad within Godot.
If you start with the cursor outside of the game window and move the mouse over the game window with the gamepad, then the cursor will not update until you move the cursor with the mouse (then the gamepad movement and cursors work as expected). If you move outside the game window and then back into the game window again, then the same behavior is seen. This has only been tested on a Mac (M1) so for.