> Are you referring to removing a live wire from the relay and touching it to the other live wire?
> Or are you referring to the 5V and signal pins coming from the arduino.
> I tested the cable that goes to the light without the relay, and the light lights up. So, there must be some wiring issue with the relay itself.
If you are sure enough about the cable, fixture, and bulb that you want to proceed as if they were proven good then you must conclude that your relay is faulty and replace it.