If your primer pump feels mushy and does not seem to be pumping fuel it probably has dirt under one or both of the check valves or they may be bent. The check valves are thin spring steel flappers that seat against the aluminum pump head housing. Unfortunately the check valves are hidden inside the crimped on top cover. This post will explain how to remove the top cover, fix the valves and replace the cover.
First you need to grind the crimped flange off of the formed sheet metal cover. Then you will need to remove the two flapper valves and clean and or straighten them so they can seal to the aluminum casting.
Then, you will need to reassemble this thing. This will require a clamping method. I made some aluminum rings which work very well and can be undone whenever the pump might need service again. Making the rings will require a lathe and a drill press. Not much else. Might be able to do it with hand tools.
Here is a photo of the clamping rings:
And here is a scan of my sketch showing how to make the rings:
And here is a photo showing the repaired pump with clamping rings:
If I ever do another of these repairs I will take some photos of the interior to add here, especially of the bent or dirty flapper valves.
It is worthy to note that the valves are delicate and may well get bent by applying compressed air to the pump. If you need to blow out your lines try to stay away from the pump.
Paul