iMac's Time Machine backup fails due to unavailable files when screen locks

Give this a try: boot into Safe Mode according to How to use safe mode on your Mac, reboot normally and test to see if there's any improvement.

NOTE 1: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

NOTE 2: if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode. This makes it act as a wired keyboard as will insure a successful boot into Safe Mode.

Do you have a Silicon Mac? If you do then these are the directions:

If using a Mac with Apple silicon

1. 1. Shut down your Mac.
   2. Press and hold the power button on your Mac to turn on your Mac and load startup options. When you see the startup options window, release the power button.
   3. Select your startup disk. It's named Macintosh HD, unless you changed its name.
   4. Press and hold the Shift key, then click the “Continue in Safe Mode” button below your startup disk. 
   5. 
      
   6. 

Did you use the charging cable with your Bluetooth keyboard?

There are quite a few users who are having problems after sleeping their Macs and not just with Time Machine backups.

IMO it's best not to sleep the Mac but use the screen saver to turn off the monitor after a set time. If you leave the Mac for any length or over night then shut it down. Rebooting is quick and not an inconvenience. Also shutting down clears our system and application temporary cache and swap files keeping the Mac lean and mean (so to speak).

Just to be sure you did set the Energy settings pane to prevent automatic sleeping when display is off like it shows below, right?

Let us know how it works out for you. 🤞