This reads to me as you copied your Home folder and data to an external drive and then went to Users & Groups to change the location of your Home folder to there, yes?
You've now broken certain critical links to your Home folder. There is a proper way to setup the machine to startup and run from an external drive, but the way you've apparently done it is not the way.
What you should have done is simply install macOS on the external drive and then boot from that and migrate your user data to that drive with Migration Assistant. Guidance for that is here: Install macOS on an external storage device and use it as a startup disk - Apple Support
Here below is Apple's guidance for changing the name of your macOS user account. There is info there that I believe is relative to your situation.: Change the name of your macOS user account and home folder - Apple Support. This process is critically sensitive and one small error can make the Mac inaccessible. I believe you did that.
And I'm not sure the fix for your situation now other than to restart the Mac in recovery mode and reinstall the OS. Before you wipe the internal drive, you might be able to wipe the external drive while booted in recovery, properly install macOS on that and then boot from it to correct the damage you've done to the original OS on the mini.
How to reinstall macOS - Apple Support