rrdandrea, after right-clicking on the shortcut are you selecting delete from the context menu that opens? Or are you pressing the delete key on the keyboard? You should be clicking on delete in the menu.

If that is what you were already doing then you might need to refresh the display to make the changes show. A way to do this is to right-click on an empty area of your desktop and select Refresh from the menu that appears.

Note that you should verify that you have a shortcut on the desktop and didn't save the file to the desktop as in that case you need to move the file to a different folder if you want to keep the file.