What is the purpose of this PCB?
It seems quite a complex circuit for what it does but in essence it runs the motor measuring the drop on the filter inductor, when the current is too high it drops the relay and stop the motor. C1 and C2 gives a delay for the intervention. For how is structured I'd think that the circuit is actually designed to be powered in both polarities (forward and reverse), so a couple of transistor works in forward and the other one works in reverse, and the diodes exclude the other pair. The C1-C2 back-to-back connection show that's designed to work with a negative supply. Is that the way is connected upstream?
As for the burned capacitor, it's only purpose seems to filter the incoming supply. It should be a (quite big) non-polarized electrolytic capacitor or a huge film capacitor. If the sticking wires are from it, I'd say a big film one. The value shouldn't be critical, the bigger the better.