Apologies for the delay but I got side tracked.
The good news is that my problem has gone away. I’ll explain what I found, and by chance it worked.
I first created a new user to see if the problem existed in the new account and it did not, any icons I placed on the desktop stayed in the correct position.
Then, later in the evening whilst I was reading some other forum posts I came across a reference to “LightDM GTK+ Greeter Settings” which I thought I would take a look at as I had not used them before.
When I opened the LightDM application, the first drop down on the appearance tab is “Theme”. As I said earlier I am using BlackMate … but this LightDM dialog was set to Ambient-MATE. No idea why, I don’t think I have ever used it before. Anyway I set it to BlackMate, came out of the dialog, rebooted and that was it, my icons had stayed in the same place. I rebooted multiple times moving the icons to check that they stayed put and they did every time so I’m fairly sure it is fixed.
So maybe there is some theme corruption when two different themes are specified.
Thanks for your help.
Martin