if you are talking about the solenoid on the side of the pump, I believe its called the timing advance solenoid, just wire it into the fuel cutoff solenoid. like you said it runs better being powered the entire time. From what I understand I think it allows the pump to have somewhat dynamic injection timing, I'm not sure what system is being used to determine when and when not to give it power but I think the only reason it exists is for emissions. Maybe it gets its signal from the coolant sensors in the rad like the glow plug circuit?