I suspect that there are several problems making you can't drag and drop files from Mac to your external hard drive. They are:
1. Drag and drop faulty
If drag and drop is not working on your Mac due to the mouse connections or any others, you can't move files to the external drive, right? Check your mouse, or you can try to copy and paste the files to the external drive.
2. No permission to write to the external hard drive
If still can't copy files to the external drive on your Mac, check if you've set the external hard drive to read-only for everyone including yourself for security.
You can change permissions on Mac by right-clicking the drive and choosing "Get Info". Then, unlock with your admin password to change the external drive to Read & Write.
3. The external HD is corrupted
I think that maybe you've done some operations on your Mac during the external drive is reformatting to exFAT. So, the external drive is corrupted on Mac and you can't drag and drop files to it.
You can run Mac First Aid to repair the disk.
- Open Disk Utility.
- Select the drive on the left sidebar and click the top First Aid button.
- Confirm your operation on the pop-up window.
- Wait until First Aid finishes checking and repairing.
If it changes nothing, repeat the reformat task. During the disk erasure process, don't work on your Mac but just wait for the formatting process to be finished.