Converting an outdoor light socket into a light socket and outlet
Thankfully, since it's a 4" pancake, you're in luck.  You can attach a weatherproof extension to it, then attach a new weatherproof light fixture to the extension (since the one you have is disintegrating), and then you can use the extension ring as a splice point to tap off wires for a new outlet box -- use THWN run though a 1/2" conduit nipple for this, and a weatherproof surface mount single gang box with a matching in-use "bubble" cover for the receptacle itself.

Don't forget to turn the breaker off before making these changes!

(Linked parts are simply examples of what can be used)