It's probably what's called a Pump Control Valve, used to avoid water hammer. There is a small solenoid valve that controls feed water from the high pressure side (coming from the pump) to a diaphragm connected to a piston like mechanism attached to the valve flapper. So when you turn on the pump, the solenoid closes and the pressure from the pump expands the diaphragm, which slowly forces the flapper open and avoids the surge of water. There is a limit switch on the flapper that only closes when the flapper is fully open, and that limit switch is in the starter hold-in circuit to keep the motor running. A timer in the circuit holds it in until the limit switch activates. If the valve doesn't finish opening before the timer times out, the pump shuts down to avoid cavitation from water not flowing.
When you want to stop the pump, your control system gets the stop command, but the pump stays running and the solenoid for the diagram opens, letting the diaphragm relax and slowly close again, avoiding water hammer. Water hammer is the kinetic energy of the column of moving water getting suddenly trapped and turning into a shock wave because water can't compress. So by slowly closing the valve the energy is dissipated over time and not trapped. Once the valve is fully closed, another limit switch that was holding the starter coil closed will open, dropping out the starter and turning the motor off.
