I would not piggyback wiring as you risk overloading the circuit that provides the left hand wiring. If you have no live at back right hand side lamp you need to trace wiring back. I would think the wire has broken from control box to light, do a continuity check with light switch ON to earth this would show you whether it is broken or not.

As for hazards sounds like relay has broken, this uses a separate relay to the indicators, sometimes tapping the relay can free the contacts.

Martin