I just tested this on my Macbook Pro with the same version of Bridge 2021 and MacOS 10.15.7. I had no problems copying and moving files using Bridge. Could you add additional information about any behaviors you are seeing?
Somethings I would do to troubleshoot....
1. Check drive space on my external and internal drive.
2. Check permissions of the folder(s) I'm copying and moving files into.
3. Restart my computer.
4. Try moving files to a different external hard drive to see if that works... could be a drive issue.
Go to your System Preferences and select Security and Privacy (found in the top section). Now click on the Privacy setting (last tab). Now click on the lock on the bottom left and provide your Administrator code. Now select the Full Disk Access option and click on the "+" symbol. This will bring up the standard Mac Open window and let you add any application you want or need to have full access. Look for Bridge, select it, it will now appear in that window and make sure it's checked. Next go to the Files and Folders option and do the same thing as above (although there's nothing to check, just add. Close everything down and see if that makes it all work. [Note: I did these screenshots when I was using Mojave so they may look a bit different than what you have now with Big Sur but the process is just the same.]
Please let me know
[Note: these images are from an older OS, the process is the same.]
I seem to have solved the problem by going down the route of Full Disk access option for bridge and photoshop as advised above. I also had a permission problem with one of my hard drives which I have sorted now.
So good to go ! Thanks for your suggestions - relieved to have it sorted .....