Hello _L_J_,
You can use apps on the external monitor with Stage Manager, as you have a compatible iPad. Check out those details in the article below:
Move, resize, and organize windows with Stage Manager on iPad - Apple Support
Turn Stage Manager on or off
With your iPad in landscape orientation, open Control Center, then tap
.
The window you’re working in is positioned prominently in the center of the screen, at a size that makes it easy to work with. All the other apps move left and form a list of recent apps along the left side of the screen.
After Stage Manager is turned on, you can drag and arrange the external display:
Move an app to an external display
If your iPad (supported models) is connected to an external display, you can work across both screens—just drag apps and windows between the screens and arrange them in a way that works best for you.
To move apps between displays, do any of the following:
•Drag from the top of the app window to move it from one display to another.
•Drag the icon of the app you want to move.
Let us know if that helps.
Kind regards.