Steve.
I am appalled at what your pictures show. That green fuse should not be loose, but then neither should there be a wire stuffed into the fuse contacts. That's the hallmark of a prize cowboy who deserves to be shot for putting your car, and possibly you with it, at risk. For goodness sake get that corrected quickly. If you can not do it yourself, engage a professional.
As for the cigarette lighter socket, the problem could a loose fuse, loose connections, or a defective lighter (they do burn out), but given the shoddy work done to the fuse board it could be anything. At the moment I'd say a faulty cigarette lighter is the least of your problems.