You could try this (worked with older Windows): Go to your user folder, and then folder "desktop". You will now see your desktop as a regular folder. Then, go to top menu, folder options, "View" tab (the one in the middle of that pop-up), and configure it as wished. In this case, forcing not showing system neither hidden files in this specific folder. (Don't hit "apply to all folders" button.)
Provided you had previously set to show system and hidden files for all folders (actually clicking in the mentioned "apply to all folders" button) in, for example, the main disk folder (that is, C:) so that all folders would show those files... Doing this after my first explained step (sorry for the mess) should keep that configuration only for the desktop folder.
desktop.ini
files on the desktop? They are really only used to provide filenames and icons for system shortcuts (eg if you copy a link to Solitaire or Notepad to the desktop). However, the shortcuts work and look just fine without them.