There should be tiny spring loaded valves at both the inlet side, and the outlet side of that pump. The tiny rubber piece you have shown MUST be inserted into the spring portion of the valve in order to build pressure.
There should be a rubber piece in the other end too. If those rubber pieces are fallen out, or displaced, other than being pressed into the center of the spring valve, the pump WILL NOT build proper fuel pressure out to your fuel lines.
There are pics in another thread on here somewhere showing exactly what that little rubber part should look like when properly placed into the spring part of the pressure valves at each end of your pump.
With your rubber parts fallen out of place, you should have been hearing that pump doing a lot of pumping (clicking) each time you turned on the key.