2 Answers

You might want to experiment starting the mower with different choke and throttle positions. Also, if the manufacturer doesn't warn against it, you can try running premium gasoline, which occasionally helps. You should be able to hear whether or not the primer is producing a gurgling type fluid sound if it is effectively operating. Perhaps you might consider priming it a little more. Myself, I often lightly spray ether into air intake with the choke open, rather than removing the spark plug. One thing you might want to investigate also, is whether or not the diaphram is pliable and not torn, as this will also affect how the engine draws fuel into the carbuerator. Also make sure your air filter is clean, as it needs to be able to effectively draw air through it while you have the low revolutions involved in starting the engine.

If you do not get gas when you pump the bulb you have a  source problem or a leaky bulb.  Source problem I have found is the plastic tube which is usually fastened very securely to  the carb base.  But if dropped off into the tank you'll have  to reinstall it.  If the tube will not stay fastened  you will need a new carb body.