Pump not pumping

It's not a one way valve, it's a water strainer. It has a barrel with a fitting on each end, the upper leads into the barrel, the lower has a strainer tube attached. When it's plugging up, it seems like a one-way valve because when you try to force water into the top, it pushes the lint and junk against the strainer screen, blocking it. When you try to force water in from the bottom, the water pushes the lint away from the screen, allowing a reduced flow. Some have an arrow; it's to indicate flow direction to keep the lint on the outside of the screen; the inside, being mush smaller than the barrel, would plug up much sooner. Before you replace the strainer, clean the lint & stuff out of the small pump reservoir (above the pump). If you don't, the strainer will plug up quickly again, if it can even get water to it.

I have bought a used replacement pump and barrel. First I swapped the barrel, still did not fix it. I replaced the pump, used pumps show up on ebay every once in a while. The new used pump worked great.

Place the old pump on a table with the two silver screws to the right, and I opened up the old pump, very carefully unscrew the two screws. Twisted it to get it off the nipples, and pulled it apart. If there are 2 springs inside the long chamber to the left you have a broken spring. The long little spring should be connected, through the shaft to a cap to the right. Mine spring broke off at the cap and the little piece was blocking the hole. The cap is fine, I need to find that spring. The spring I am talking about is about 1 inch long and fatter on one end than the other.

I have the same problem and I am glad there is a solution but can you tell me if I have to take the unit apart in order to access this valve acting as a filter?

It's acting as a filter. You have to clean it periodically.