Try opening Finder and Go to ~/Library/Preferences, locate and delete the com.apple.systempreferences.plist file, restart your computer, and then try opening the App you need to add and hopefully it will send you to the Privacy Settings to enable it. (Detailed Steps for Deleting .plist files: https://discussions.apple.com/docs/DOC-4301)
I work as a Help Desk Technician and one person just reported a similar issue and the steps above worked.
They had no option to enable the application they needed to use in the Privacy settings.
I noticed the "+" button was missing on my machine as well (Catalina 10.15.4).
From my experience if System preferences is acting wonky, deleting the .plist file may fix the issue. Trying to create a new user account is another good troubleshooting step to test if it's a profile issue. If no resolve then I might try wiping and reinstalling MacOS and start from scratch.