Lately it seems if you let your study desk empty entirely your character will not automatically start studying after you refill it. So before refilling the desk manually start your character studying a few items (right click>study) then fill the desk, it should work normally again (unless you let it empty).
If you right click your own desk the only option is "open", if you right click any other desk you have the "open" option and the "take possession" option. You can only possess one desk, so if you take possession of any other you will stop studying from your own desk until you take possession of it again.