Ah Ha! Having trawled the net a bit more, all of a sudden I understand. I shouldn't be trying to run Software Center at all - that's for Ubuntu-based systems. Since losing track of the facility to scan software sources many months ago I had forgotten that the Mint Debian version that was on my laptop was called Software Manager.
I had tried installing Software Center on my xfce system using synaptic but it refused to run.
What I needed to do was use synaptic to install debianinstall . That did the trick. Rebooting then re-established the tool in the correct place on the menu.
Ancient but excellent Sony laptop now running Mint Debian Mate. Spends much of it's time as a wireless controller for a Logitech Squeezebox.