Is there a way on an iPad running iOS 16 to recover deleted voice recordings?

Do you have an iCloud or iTunes backup of your iPad that pre-existed deletion of the files - that has not been refreshed since the files were deleted from the iPad? If so, in restoring the backup, you may be able to recover the deleted files:

Restore all content to iPad from a backup - Apple Support

Be aware that in restoring the backup, all files on the device will be replaced by those of the backup. Any data that has been added or altered since the backup was last refreshed will be lost.

Just to be sure, you have actually checked the Voice Memos Recently Deleted folder? When deleted, recordings are initially moved to the Recently Deleted folder - accessible from the Voice Memos sidebar...

To expose or hide the sidebar, simply tap the Sidebar button at top-left of the App window.

Within the sidebar, you will see all folders that you have created - along with the Recently Deleted folder if any deleted files are present.

Deleted files may be automatically removed after 30 days.