If you think the problem may be with explorer.exe, why not close it and then restart it?
To close and restart Explorer.exe first go into Task Manager, Processes, highlight 'explorer.exe' and click on End Process. (At this point, you may temporarily lose your Desktop.) Then, to restart explorer.exe go back into Task Manager, and in File section, select New Task (Run). Type explorer.exe (or C:\Windows\explorer.exe) in the Run box (it may be there already). Below the command, it should say: This task will be run with Administrator privileges. Click Okay.
If this doesn't solve the problem, you might try pinning the internet shortcut to the Start Menu, and then right-clicking on it and selecting: Send a shortcut to the Desktop. Once the new shortcut is successfully on your Desktop, you can delete the pinned Start Menu shortcut, if you wish.