I can't add folder or select anything on my desktop

Select the Desktop folder and type ⌘+i to bring up its Info pane. Go to the Sharing & Permissions section at the bottom to see what Privileges you have.

If it's not Read & Write then click on the padlock, enter you Admin password and change it to Read & Write. Then click on the circle with the three dots in the center button and select Apply to Enclosed Items:

What happens, when you click on the empty Desktop? Your screenshot is showing, that Safari is the active application. Is clicking on the Desktop bring the Finder to the foreground?