It's a voltage drop due to current draw.
With the fog light in the on position (and headlamps if you need them on to activate the fog??) run a wire from the chassis somewhere to the negative side of the lamp, if it lights up you know it's a bad earth. If it don't then you know it's the poss side at fault. If the lamp didn't light up then I'd cut an inch off the red/blue? wire at the lamp end,reconnect and try again. If that didn't work then see if the switch is at fault by pulling the plug off the bridging the two contacts, if that lights the lamp the switch is at fault. If that fails to sort it, run a long wire from the other wire on switch plug red/orange? directly to the red/blue terminal on the fog lamp. If that works then the wire if broke between the switch and lamp.
Jason