I use Borg to compress, encrypt, and deduplicate my backups, and it has been working really well. Recently, I was trying to retrieve some files and got the following error messages:

...
cp: cannot open '/path/to/borg/mount/file.txt' for reading: Permission denied
...

After reading documentation, the solution I found for this issue is to provide mount options to Borg, setting the permission of the files mounted.

borg mount -o uid=$UID,umask=077 /mnt/drive/borg::my-backup borg-mnt

This resolved the issues.