Thanks for that very prompt reply Biggsy. If possible I prefer not to add more software in order to remove something else, so I followed your tip to check the Task Scheduler but there didn't appear to be anything relevant to Magic Desktop. There is no sign of it in Startup either. So ...
I checked C:\ProgramData and found a folder
Easybits Magic Desktop for HP
which I renamed as
Easybits Magic Desktop for HP_Renamed20131220
Shortly after I did this I got the exact same update notification as shown in my OP
A bit of googling suggested these update notices come from an EasyBits specific program called
C:\ProgramData\Easybits Magic Desktop for HP_Renamed20131220\mdhpSUN which I renamed
C:\ProgramData\Easybits Magic Desktop for HP_Renamed20131220\mdhpSUN_renamed20131220
I'm assuming that renaming the folder and the program will make them invisible to whatever is using them. If that works, and there are no consequential problems, I'll go back and delete them.
If none of that works I'll undo the renaming and adopt your suggested plan B and download CCleaner or something similar.
Doesn't look like there will be an instant answer, but to close this thread I'll report the final outcome here.
Thanks very much again - its been a learning experience