You may have a corrupted 'com.apple.finder.plist' file.
To rename / delete the 'com.apple.finder.plist' file ...
01. Select 'Finder's 'Go, Go to Folder' menu item ('Command <shift> G'). If no folder window's currently exist - a new 'Go to Folder' search window will appear; otherwise, a sheet will drop down from the front most folder window - stating 'Go to the folder:'.
02. Enter ...
~/Library/Preferences/
... and ciick on the 'Go' button.
03. Locate and, rename the 'com.apple.finder.plist' file (to 'com.apple.finder.plist_', or something else) or drag the 'com.apple.finder.plist' file to the 'Desktop' or 'Dock' based 'Trash can'.
04. Press and keep pressed the '<option>' key as you click on the 'Dock' based 'Finder' icon, a sub-menu list will appear.
05. With the '<option>' key still pressed, select the 'Relaunch' menu item. You can now release the '<option>' key.
If the 'com.apple.finder.plist' file was the problem, the 'Desktop' items icons should now reappear.