Can't change photos in Preview.app – permission problems only in ~/Desktop
Right click on the Desktop icon in finder, show info. Is the the folder selected as 'Locked' for some reason (little padlock bottom left corner of icon in Info window)?

Expand the 'Sharing and Permissions' section, should say 'custom access' and the privilege should show Read &amp; Write.

Click the little padlock bottom right of Info window, enter admin details and click the little cog, and select 'Apply to closed items'.

Could also open Disk Utility, select the drive, click 'First Aid' and check the volume for errors. This last may not have any effect, and should come back showing as all good. The 'Repair disk permissions' will be completed automatically whenever you run any software updates and does not require you to do it manually any more.