I have had similar occurrences, though I can't recall the specific cure. Here are some basic troubleshooting steps I usually take:
1) Since the mower does not want to run even when primed, I doubt that the carb is the main issue. If it was the carb, you would at least be able to run the mower by constantly pressing the prime button when it starts. This is good when the carb bowl has no gas in it.2) Check the spark plug. What color is it? If it is black, then you have a fouling issue. Is it wet? If so, then it means that you are getting fuel, but no ignition. Clean the plug (or replace it). Make sure the plug is dry when replaced.
3) Check the air filter. Make sure that it is clean. A dirty filter will cause your mower to run rich and foul the plug. In extreme circumstances, a clean plug can foul in five or ten minutes.4) If you are still not having any luck, remove the spark plug and reconnect it to the plug wire. Have the metal part of the plug touch the engine and pull the start cord. Do you see a spark? If yes, then you have a fuel delivery issue. If not, it could be the plug or the coil. Keep working with the above steps to see if it will run. If you still can't get it going, take it to a small-engine mechanic.