Try this, from here:
http://help.lockergnome.com/vista/Desktop-Icons-delete-ftopict65186.html.
The Desktop is actually a glorified interface for two folders:
C:\Users\(Your Name)\Desktop
and
C:\Users\Public\Desktop (which is hidden).
If you have the same file in each folder, they will both show up on your
Desktop. The Public Desktop should be owned by the Administrator and
C:\Users\(Your Name)\Desktop should be owned by the user.
If you are the Administrator, can you navigate to these folders using
Windows Explorer and delete them there? If your context menu is gone (which
is another issue), can you select one, and use Organize > Delete? If the
files aren't in those folders, but they are on your desktop, try going to
Control Panel > Personalization and then click on Display Settings. Change
the Colors to a lower setting (e.g., from Highest to Medium). Then do the
same again, and reset Colors back to the way it was originally.
If that doesn't work, try deleting the (hidden) IconCache.db file under
C:\Users\(Your Name)\AppData\Local\ and restart your computer. A new one
will be built automatically.