How to delete a desktop shortcut without deleting the actual programme - W10

Solution 1:

Deleting a shortcut never deletes the application associated with that shortcut. If you look closely at the wording of your second screen shot, you'll notice it's asking

...permission to delete this file

Confusingly the location mentioned in the dialog is the folder name that contains the shortcut's target. You can confirm this by inspecting the shortcut's properties and observe that this folder name is present in the path of the shortcut's target. Neither this folder nor its target file are deleted when the shortcut is removed.

The reason it requires administrative permissions is because the shortcut is located in the Public Desktop folder which is shared by all users of the computer. Due to that folder's permissions you must have admin rights to modify its contents. This prevents non-admin users from making changes that would affect other users of the computer.