It sure can be frustrating when you have to pull your lawn mower pull cord more than 2 times to get your mower to start. You pull and pull and nothing until finally it fires back to life. A normal push lawn mower should start in 2 pulls. If it doesn’t, then it is having a problem and that problem needs to be discovered.
It could be something simple such as a bad spark plug or that you might not have any gas in your lawn mower. It could also mean that your carburetor is getting plugged up and needs to be cleaned. Stale gas can also sit at the bottom of the float bowl and not have enough volatility to light easily.
The timing and valve adjustment could be incorrect as well. Check the clearances on both the intake and the exhaust valves to be sure that they are within specifications. Also check the flywheel key to be sure that it is not sheared. A partially sheared flywheel key puts the engine out of time and will make it harder to start.
By systematically eliminating what it is not, you can resolve the problem with your push mower taking a lot of pulls to start.