The relay is probably a double pole relay, that means it would have both power wires and both pump wires connected to it. The pump and power wires will be the larger, heavier wires. The control voltage for the relay is usually a smaller gauge or even could be very small low voltage wire. When the relay is energized by the control wires, it closes and simply connects the two power wires to the two pump wires like a big heavy duty switch.
To bypass the relay you would need to disconnect all four wires from the relay, then connect one of the pump wires to one of the power wires using a wire nut. Then connect the other pump wire to the other power wire using another wire nut. I'm assuming this is 240v, if so polarity does not matter.
The pump should start as soon as you flip the breaker back on.
This is 240v and very capable of severe injury or even death. You should ALWAYS test using a reliable tester to be absolutely certain the power is off before you touch anything. If you are not completely confident in your ability, then call in a pro. No need to wait for the pool company, any qualified electrician will likely be able to repair the relay or at least get your pump running until parts can be obtained.