I don't no about the fog light but there are 2 in the box i mentioned i have swapped them around the car started immediately and gave no trouble of course the bad relay I replaced any car dismantlers would have them.
There is a relay beside the battery in a box the first controls the radiator fan the other one could be your problem they tend to over heat or give intermittent problems its in a long rectangle box swap it out for a new one easy to get there blue in colour .