Stage Manager and full external monitor support is currently in its first production release - and it is perhaps inevitable that some performance issues or functional issues may be encountered.
A forced-restart will often resolve an isolated software issue - and this may be a worthwhile initial step:
Force restart iPad (models with Face ID or with Touch ID in the top button)
If your iPad doesn’t have the Home button, do the following:
- Press and quickly release the volume button nearest to the top button.
- Press and quickly release the volume button farthest from the top button.
- Press and hold the top button.
- When the Apple logo appears, release the top button.

Force restart iPad (models with the Home button)
Press and hold the top button and the Home button at the same time. When the Apple logo appears, release both buttons.

If problems persist, you might be well advised to seek assistance directly from Apple Support. They may be aware of issues - and may capture issues that they see reported to them.
You can contact the Support Team using the Support link at top-right or bottom-left of this page. Alternatively and perhaps easier, you can initiate and manage your support cases using the excellent Apple Support App. If not already installed on your iPad, the App can be downloaded from the App Store:
https://apps.apple.com/gb/app/apple-support/id1130498044
For repeatable issues, you might also consider submitting a Bug Report via Apple’s Product Feedback portal. For iPad and iPadOS:
Feedback - iPad - Apple