That might be the start capacitor, though usually they are permanently wired so would discharge through the windings.
It sounds like you've taken the electrical cover off the pump but not the pump head end (the bit that does the fluid movement). This should be dismantled to check for blockages, and this is the place you have to be careful not to lose the little pump vanes.
If it's the pump I think it is (got any photos?) then it'll look like this inside:
The little vanes get flung out by centrifugal force, they just sit freely in the slots.