Your code is basically okay, assuming that you have appropriate rights on all of the computers you’re connecting to (access to the c$ share, permission to remove files from users’ desktops, etc. Basically, Administrator), with one exception. There is no C:\Users\Desktop folder. Instead, you want to remove the contents of the Desktop folder from each user profile found in C:\Users (assuming that user profile directories are their default values and haven’t been redirected.) Try this:
Remove-Item -Path "\\$comp\c$\Users\*\desktop\*" -WhatIf
I added the -WhatIf switch because this is a destructive command, and you might want to make sure you like what it’s doing first. You might also want to add a -Recurse switch to the command, if there are folders on these desktops that you want to clear out.