My operating system upgrade from Mojave to Big Sur has rendered CD's unreadable somehow. How can I determine if a CD is HFS or what, and is there a work around or fix for this problem?

The issue is mentioned in your initial post. Mojave was the last version of Apple's OS that could read the original HFS format. After that, disks must be at least HFS+ (Mac OS Extended).

The easiest solution would be to use any other Mac running Mojave or older, where you could copy all of this older data off the CDs and put them on a flash drive, or other media using HFS+ or APFS.

If you don't have access to another Mac, use Disk Utility to create a new partition. Install Mojave or earlier to that and use that OS just long enough to transfer your data to a newer disk format. You can then boot back into Big Sur and dump the Mojave partition.

This is why you need a second source for backup.

DVD/CD drives can get dirty.

Discs can wear out with age.

If disc was not stored in a jewelcase, it can easily get dirty.

If disc had any writing on the label side, its reflectivity can get messed up on the other side, plus the substrait can eventually wear away.

If there was a sticky label applied, it can make the disc unbalanced.

I always recommend keeping discs in jewelcase with writing on the outside of the case, or in a paper sheet that is included with the case to identify the contents.

You can try to create a downgrade partition to see if there is something wrong with your DVD software on you computer:

How do I downgrade my Mac OS X operating … - Apple Community

If that doesn't work, see if getting an external DVD drive from http://www.macsales.com/ helps.

If not, you may want to take the disc to a data recovery specialist.