Thanks. Forget about the "thumbnail" terminology for now... it was just confusing me.
sciencemagic wrote:
Hi John Galt,
You said, "Click anywhere on that Mac's Desktop background, then choose File > New Folder. An empty folder with title untitled folder should appear on the Desktop. Does it? When I open "new folder" a box appears for Finder? I then get the message "The operation can’t be completed because you don’t have the necessary permission."
That is not right. It means the Desktop permissions have changed for reasons unknown, which suggests other unknown changes may have affected that Mac.
The best way to describe how to change your Desktop folder permissions is to read Apple's Support document on the subject, here: Change permissions for files, folders, or disks on Mac - Apple Support
To change permissions for your Home Folder's Desktop folder, click anywhere on the Desktop background and then choose File > Get Info. The Get Info window should closely resemble the following:

Where it indicates "Read & Write" in the above example, yours is probably different. What does yours indicate?
- If it is not "Read & Write" you will not be able to alter anything on the Desktop.
- If "You can only read" appears under Sharing & Permissions, same problem. You won't be able to change anything.
To change the Desktop Permissions, follow Old Toad's instructions: click the padlock icon (lower right), authenticate with your usual login password, and change the dropdown menu to match the example indicated above. If it asks to "apply to enclosed items" say yes.

This may not be a permanent fix, or it might not work at all. Permissions don't get changed on their own, which is the reason I asked about that "HopToDesk" concern. If someone or some thing altered your Mac in an inept or malicious manner then greater concerns exist. If it works though, and everything else appears to work as it should, then I'd put the matter behind you.
Let us know what you determine.