Sounds like your car runs out of fuel. Its either injectors dont have power or the circuit opening relay aint working correctly (assuming you have wired it with one).
Im more pointing to the latter. You probably have the start signal hooked up to the relay so that when it cranks the fuel pump works, but you have an issue with pin FC (green/red wire) not being earthed when car is running. You either need to connect that wire up to the ecu OR check that the ecu is working correctly. Pin FC on the ecu has been known to fail on some occasions, and you may need another ecu.
A temporary fix would be to earth the pin on the relay, but only for the short term (pump will work as soon as ignition is on).