Cannot write on TimeMachine data disk - told "Don't have enough permissions to write to '/Volumes' - files say (unknown) read&write
So I assume you don't keep anything valuable in there (those files will NOT be backed up). I also suppose you connect to the TC share using SMB or AFP or the Finder windows sidebar: just check the credentials with which you are connecting (clic on TC in side bar, clic "Connect as..." in top bar). I never had any luck trying to change user permissions on a share from another machine. I believe you should check AirPort Utility/File Sharing/Secure Shared Disks/With accounts looking for user credentials, following these instructions: https://support.apple.com/en-us/HT202254. Hope this helps.