Is this a car you've recently purchased or one you've had for a while and it's just started having troubles? It appears to be a K-jet 240 (what year?) If it's been sitting without being used, there are a number of potential issues, such as poor/old fuel, gummed up fuel distributor, gummed up fuel injectors, clogged fuel filter, rust in fuel tank, that could cause it to be low on power and stop running then re-start after it sits for a while...especially rust in fuel tank that blocks the filter "sock" in the tank as when you shut the car off, the rust can disperse away from the in-tank filter over time, then when you start driving again the rust gets sucked back to the filter and it blocks up again causing the car to starve for fuel.
I'm not sure why pulling the spark plugs out would do anything?? Unless it's running extremely rich and the plugs are getting wet with fuel? What do the spark plugs look like? Are they black and sooty, wet, oily, dry and white? That could help diagnosis.
You should be able to purchase the vacuum & fuel hose from any auto parts store or even Clark Rubber. Just take the old piece in and get them to give you the same size. Is the one-way (check valve) bad? You should be able to blow air through if you blow on the white side, but no air should blow through if you blow on the black side.
I don't recall the fuel pump relay being wrapped in the black goop (maybe the wiper delay relay?) I guess the fuel pump relay may have a layer of white or yellow foam wrapped around it that is stuck on by a thin brown double-sided tape. Just make sure what you think is the fuel pump relay is indeed the fuel pump relay. Usually the car would die totally if the pump relay overheats or is bad...it wouldn't just run slowly. Another possible problem would be you're not getting enough fuel due to a failing in-tank or main fuel pump.
Hope that helps - hard to know what's wrong without knowing more about the car and its history.
