After discussing the problem with Apple support., I have a permanent fix that works for me. It seems the desktop.plst is not actually involved. Firstly I was asked to repair the permissions on the Mini's hard drive. To do this you need to shut down the computer and when restarting using the On/Off switch at the same time keep the Command and R keys pressed on the keyboard until the OS X Utilities screen comes up. This may take a few seconds and you need to keep the two keys pressed until it does.
Select the Disc Utilities option and then the Mac HD. Click on the repair permissions. I am not entirely sure this is part of the solution, but I am relaying the sequence of actions that led to the problem being fixed. Whilst in the disc utilities I also checked the external drive that my desktop photo is stored on. It was fine.
After restarting the computer I restored the correct desktop and openned Finder and used the Go menu to access the Library as before. Then I was told to open the Applications Support folder and then look for the Dock folder and open that. There are two files, one of which is "desktoppicture.db". Move this file to the trash and securely delete it. Restart the computer. A new "desltoppicture.db" will be created on startup.
Finally, reset your desktop picture again using System Preferences. It has from that time on worked fine. I hope this works for everyone else.
Thanks to Martin and Thomas in Apple support for talking me through it successfully first time.