Should outdoor lamp post be on GFCI

This style center screw extends into the outlet: it's possible both parts need replacing. Pull on the cover to add tension when removing the stripped screw.

Turn off the juice before taking the cover off.

Your new 'in service' cover comes with a foam gasket: do not use caulk.

Your existing box is probably fine, but when things are open evaluate if caulking is needed around the box (not the cover, the wall box).

But this is outdoors, right? That outlet should be 'GFCI' protected. And for that I recommend you hire a professional, or find a handy neighbor/friend/relative. Typically a GFCI outlet is placed right in the box, and it uses one large square opening, not the style shown in the picture. Other configurations are possible.

Surface mount a weatherproof 4/O box extension over the top of your existing box and feed the wires through the back. Seal around the box where it meets the wall so water cannot get behind it.

Then install a weatherproof conversion cover, they are fairly new and I have only seen them on the internet. Then install a weatherproof receptacle cover and you are done!