Water leaking from a hole under the engine near the water pump when the engine is runnig. dose this mean the water pump gasket has blown?

check the engine for where the water is leaking. I had an engine leaking at the back side and it was corroded
Welsh plugs in the engine. If it was the water pump water it would be leaking out the water pump. Check Hoses with engine hot . Also check exhaust pipe is not blowing steam out from a blown head gasket.

the water can get into the sump oil from pin holes in the timing case behind the water pump impeller
if you have water in the sump oil it will be a dirty grey color and the filler cap will have water droplets under it
If the oil is normal color then the loss could be from rusted welsh plugs
pressure test the cooling system to check for leaks

The water pumps have a small hole in or near the bottom. When the engine is cold, it will seal the hole but when it warms up, it will expand the seal and leak. You should be able to see coolant coming out from under the vehicle or using a mirror. If there is coolant coming out, replace the pump. Changing the gasket is not going to help if the internal seal in the pump is bad. Hope this helps.

If you have coolant leaking out of the muffler the head gasket is most likely blown. The head gasket has passageways through it for coolant water, exhaust, intake and engine oil. Depending on which location the gasket has ruptured you can mix fluids or gasses in several combinations. The only way coolant can get to the exhaust is through a cracked head or blown head gasket. If the head is cracked and another head is installed a new head gasket is installed because reusing a used gasket is very risky. The only to be sure what is broken is to take the heads off. If only the head gasket is blown the head is usually warped and should at least be milled if not replaced. I hope this helps.

If its just condensation in the oil cap or dipstick, to some degree its normal. However if you have water in the oil and its very creamy in texture, its probably a head gasket leak, or cracked head. Its important to get it repaired soon, or total engine bearing failure will result.

Yes its likely most water pumps have a small hole on the under side of them called a weep hole. Its there to let you know that the seal is starting to leak and the pump needs to be replaced before it completely fails say on the highway and dose a lot of damage to the engine If its coming out the bottom center of the pump shaft and just driping a little then its time to replace the pump you should be able to feel the hole with your finger

Is the antifreeze leaking out of a small hole on the water pump housing just behind the pulley?
You need to know exactly where the leak is coming from. If the pump is not leaking then there is another location.
Generally pumps either work or leak. They do not slow down and overheat the engine.
Where did the smoke come from? Under the hood or out the exhaust pipe? If you blew a head gasket and the exhaust was blowing steam that would cause overheating. The original leak under the engine would cause overheating that then will cause other problems like blown head gaskets.
Did you use stop leak or some quick fix? what did you do to stop the original leak?

CHECK FOR LEAKING WATER PUMP WEEP HOLE OR GASKET.ALSO CHECK WATER PUMP HOSE FOR LEAKS.IF ALL LOOKS GOOD.YOU COULD HAVE PINHOLE LEAKS IN RADIATOR CORE LEAKING ON FRONT OF ENGINE OR INTAKE MANIFOLD GASKET LEAKING COOLANT.

Exhaust and coolant problems are two separate things. It's possible the donut that connects your exhaust manifold to the pipe came loose or the exhaust manifold or pipe has a hole from rust. You obviously still have a coolant leak somewhere. Start the engine and look under the car to see if you can see coolant dripping. It's either coming from one of your hoses, thermostat, water pump or coolant lines that run from your radiator to your transmission (if you have an automatic). There's a good chances are if it's still leaking the new water pump may not have been installed correctly. Also check your oil to make sure it's not milky white. That means your oil is mixing with the coolant from a blown head gasket. That can happen if the engine overheats severely. Head gasket repair is very expensive.

have coolant system pressure test.what they do is pressurize your coolant.about 14 psi look in radiator for bubbles if so you have head gasket leak. or they drain oil out leave drain plug out to see if coolant come out crank case.that also means leaking head gasket or intake gasket.they will look for leaks around radiator +all hoses.look at water pump gasket weep hole.for leaks also check engine core plugs for leaks.they will make sure you have 50% coolant and 50% water .if you have too much water.it will evaporate away.