I don't understand that as I have Lightroom 4 that was migrated over from an older Mac to a new Macbook Pro 16" running Catalina, and I can run Lightroom 4. There are some parts of Lightroom 4 that I know are not fully 64-bit, but those 32-bit pieces apparently only affect a few functions and the program appears to work for the functions I have tested so far.
Here is the version I am running under Catalina:
Adobe Photoshop Lightroom 4.4.1 [908273]
There is a more insidious possible cause for these pop up messages, namely that you have adware somehow installed on your computer. Running Malwarebytes might identify something like that and remove it for you. Have you installed any "Cleanup" or "Clean my Mac" type programs? They can cause such pop ups which might not be legitimate messages from the OS in that case. Or some malware could have embedded itself in your Adobe software.
So I suggest: (a) checking your version of Lightroom 4, making sure it is the latest one, and (b) using a safe program like Malwarebytes or Etrecheck to scan your computer for malware and remove it if any is found.
By the way, Lightroom 4 is long been unsupported by Adobe so some unpredictable results may occur when using it. Adobe now offers a fairly low cost Adobe CC package for photography that includes Photoshop and Lightroom both, and with that you get the very latest versions that are compatible and have the latest features and bug fixes.