after shut off because you're in the early stages of seal failure. Seals
fail from running w/o water
, bearing wobble or age. After the engine has run, it usually heats the coolant above normal because the coolant isn't circulating. With heat in a closed system the pressure rises. The increased pressure is relieving itself thru the seal.
Any pump that was built can be rebuilt. The problem is getting the parts
. You need a new seal. I don't know your engine, but usually the pulley has to be pressed off, and behind that is some kind retaining ring that holds a bearing in place. Or maybe the bearing is pressed in.
Better auto parts
stores can do the pressing and maybe find the seal and bearings. But it's not always cheaper than buying
a new one. Try ebay. That's where I start for parts. Sometimes Amazon. Last choice is OEM or marine