Depending on your carb, sometimes you can fix the "initial dribble" problem by adjusting the pump rod so the pump starts off further down in the bore. It may even be out of spec with respect to that adjustment, or your float level may be too low. But it should be easy enough to measure where your pump is sitting at right now (in case you need to change it back) and bend the rod so it sits another few millimeters further down at rest.