Many are reporting difficulties with external display support after updating to iPadOS 16.1. Be aware that Stage Manager does not yet support external displays. With Stage Manager disabled, standard screen mirroring should still be available via AirPlay or HDMI.
If you continue to experience difficulties, you would be best advised to seek assistance directly from from Apple Support. You can contact the Support Team using the Get Support link at top-right of this page. Alternatively, 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
You might also visit the Genius Bar at your local Apple Store or Apple Authorised Service Provider (AASP) - where the technicians will be able to test and assess your iPad and HDMI Adapter - and you’ll be able to compare your results with display models iPad that will invariably have been updated to iPadOS 16.1.
Genius Bar Reservation and Apple Support Options - Apple
If a bug is confirmed, Apple invite submission of bug reports to Apple using their Product Feedback portal - this being important, as resources to address the issue are unlikely to be assigned unless or until a sufficient number of reports have been submitted. Every report counts:
Feedback - iPad - Apple
When submitting a bug report, ensure that you include details of your iPad model and installed version of iPadOS within the body text of your report.