my macbook pro can't read my external hard disk 5 TB which formated exfat

I've had this issue using Macbook's time machine - all of a sudden it won't mount my external drives -(I have a backup with Time Machine and I have it partitioned so part of it is time machine and the other is for drag and drop. I also have additional external drives that just copy movies and graphics - as well as larger files to take off the mac for extra space. I'm still on Catalina as I was discouraged from upgrading to Big Sur - but may do so as long as I can get my external hard drives to be recognized. They were EXFat formatted - and I don't want to reformat them as it would erase what I have on them and that was the whole point!! I'm on a MacBook pro late 2019 model. I've noticed there's a lot of issues with the Macbook pros not recognizing external hard drives and some of the hard drives are not that old. Has anyone found a solution?

It is best to erase (aka format) the drive using Disk Utility on the Mac. Windows may use file allocation sizes that are not compatible with a Mac. Plus you should not use any third party software on a Mac to manage a drive. macOS has native support for exFAT.

I maintain data and data history (Safety Net) via Carbon Copy Cloner. I treat my TimeMachine backups as snap shots in time of the most recent installation. I have no use for history associated with TimeMachine, so reformatting a TM drive is no big deal. TM is rather limited and seems cobbled together to me, hence my solution to use CCC. And since I have all Intel Macs, I can also create (and have tested) bootable clones.

I understand not wanting to erase history; I don't want to either. To use TM to do that is just not my chosen solution so at this point I'll yield to better minds on your problem.

I understand wanting to use a free native app to backup, but sometimes you have to open your wallet to have a workable solution over the long haul. Especially if the native app has long standing issues.

I've had the same problem. Reformatting the drive using Disk Utility always fixed this issue for me.