Having tested both my Lightning and USB-C SD Card readers, I’ve not been able to reproduce your issue. As such, the problem may be with your iPad.
As the problem occurred subsequent to the most recent iPadOS update, it might be reasonable to assume that you have a software issue - that might be resolved with full reset of iPad settings.
Before performing the reset, you would be well advised to ensure that you have a current iCloud or iTunes backup. While no data should be lost, having a current backup is a precautionary measure:
How to back up your iPhone, iPad, and iPod touch - Apple Support
To reset your iPad:
Settings > General > Transfer or Reset iPad > Reset > Reset All Settings
After reseting the iPad, you can again test your SD Card reader.
If the problem hasn’t been resolved, your next step is to perform a full reset of your iPad. This reset will erase all data and settings - after which you can restore your data from your backup. To perform the full reset:
Settings > General > Transfer or Reset iPad > Erase All Content and Settings
This support page outlines restoring your iPad from your backup:
Restore all content to iPad from a backup - Apple Support
Should the problem remain unresolved, a factory restore is recommended. This process will install a fresh copy of iPadOS 16.4.1:
Restore your iPhone, iPad, or iPod to factory settings - Apple Support
After performing the factory restore, you should initially set-up the iPad without restoring from your backup - and test your SD Card reader. This step will test your iPad and SD Card reader in isolation. After performing the test, hopefully demonstrating successful access to the SD Card Reader, you can restore your date and settings from your backup.