If the relay is being activated as designed, you can move past that as being suspect.
About the little circuit board. The parts on it are very dirty, difficult to tell if they are burnt. Try to clean the dirt off with a paint brush or tooth brush and then inspect for burnt-looking components.
If you are comfortable with a soldering iron, I advise to touch up every solder connection on the circuit board. Melt the solder and let solidify again. Ideally, dab a little flux before heating to help the solder flow and re-attach better. You might get lucky and make an electrical repair to a hairline solder fracture that is sometimes very difficult to see.
Good Luck
