My customer has a pool pump (120v/5amps) which is relatively new (3 weeks old). Here is the problem, the customer without checking with me first ran 12-2 UF cable on the surface to keep the pool circulating through out the winter so the pipes wouldn't freeze. However, the pump is located approximately 225 ft from the main panel. The pump was plugged in for approx 2.5 weeks.

I calculated the vd at 2.25 volts which would require number 10 wire. Now when he plugs the pump in it kicks the breaker. This circuit is not on a GFCI protected. The customer did this without checking with me also.

Here is my question: Is there a way to check the leads on the motor for a direct short. I suspect the motor has burned out.

you could use a continuity tester or a multimeter put one lead one each conductor to see if there is a short.

with the motor leads not tied to anything, You should'nt read anything on the meter from the wire to the case of the motor.
Put the tester across the two motor leads and you should read the resistance of the motor winding and not a short circuit.