1. Make sure your computer is clean. Run a full virus scan.
Recommend using Malwarebytes, Best in the business.
If you don't have it, here is a download link. Select the free edition. You can but the paid version later on if you like it that much. http://www.malwarebytes.org/antimalware/
Those Desktop icons will not show on your desktop screen unless you have them turned on. Even when turned off, they will always show in the common Desktop folder and it never present in your C: Users\(user-name)\Desktop folder