[Sirrotica]

The pump is activated by a metal arm that engages in the nylon gear under the pump housing. A broken nylon cam gear, or stiff dried up grease, spring coming off the arm, will make it malfunction, and not stop running the pump. There are a lot of small parts under that cover, many made from nylon that used to break even when the cars were under 10 years old. From your description it sound like the white cogged gear may have broken off one of the pawls that stop it after one cycle. The gear is about an inch in diameter, and last I knew wasn't available seperately.

Back when these cars were used daily drivers, repairing the pumps, and the mechanism that drives them was fairly expensive. Even after spending a good sum of money on the repairs, they sometimes still failed to work as they were designed.

The aftermarket sold washer pumps that retrofitted the troublesome GM design, and were installed inline in the washer hose. They didn't run a full cycle and empty the washer reservoir prematurely, they only ran while the washer button was depressed. GM defaulted eventually to this style of pump after decades of using the failure prone design. It cost about 1/2 as much to buy the aftermarket pump as the GM style did.

If your car is a judged original show car then you might want to refurbish the OEM pump. If it's a driver you might want to consider the rotary aftermarket style.