So you can't even select the icons at all? They act like they are part of the background?
Try this, change your wallpaper.
There is an old joke IT people do, they take a screenshot of the desktop, make it the wallpaper, and then hide the icons on the desktop. So the desktop is there, but the icons are just a picture. When you start in SafeMode, the background is not loaded, which is why you probably can't see these icons then.