The black part seems to be labeled "TDR1" so I would suspect that it is in the motor start circuit. It might be a capacitor to protect a Time Delay of some sort...

You might be able to re-solder the connection, but the "TDR1" part may already be toasted. So, I strongly suggest a replacement board. Check with the manufacturer or a trusty repair shop to see if that model any circuit board has replacement Service Bulletins.

On a similar note, I have seen such a failure in a furnace control circuit board that allowed the burner to start, but failed to start the main blower. The result was the burner stayed on and burned up the furnace, putting a lot of soot through the house, but fortunately not starting the house on fire. The circuit board in the furnace was an older model that should have been replaced under a Service Bulletin, but it had ntobeen replaced. I'm not sure who will win the lawsuit...