This is not as easy as it appears!
The image below is the proper routing for the fuel lines from the tank through the primer and carburetor.
It is so easy to assemble this backwards and pump unfiltered fuel through the carburetor. Or Assemble it so that the primer pushes fuel into the carburetor.
The primer pulls the fuel through the filter located in the tank up the inlet hose into the carburetor internal fuel pump filling up the carburetor and purging out any air. The output of the primer returns any excess fuel back into the tank via the return hose.
The center connection at the primer bulb is the output return to the tank and the off center connection at the primer bulb is the inlet from the carburetor.
If it is correct you need to perform a leak check on the fuel system. Reach into the tank and retrieve the fuel filter attached to the hose. Carefully remove the filter from the hose and with your finger over the end of the hose pump the primer. You should hold a vacuum. If the primer will not stay down air is leaking into the fuel system. Any air will disable the engine from running. The carburetor needs to be air free to function.
If air is not leaking from a hose connection and is internal to the carburetor it is likely that contamination has logged either in the needle valve or check valve.
Below is the check valve that prevents air from entering the fuel passages from the fuel jets when you prime/purge the carburetor with the primer bulb.
If the carburetor is filling with fuel when you prime it is likely that the the primer is pushing (incorrect hose routing) the fuel into the carburetor through the impulse pump and through this check valve. Correct pluming and try again.
The needle valve is located under the four screws under the diaphragm and is actuated by the diaphragm.
Cleaning the check valve is almost impossible. Most have a rubber washer that can easily be moved out of place from the use of excessive air pressure. The claim is that the check valve can be replaced. I've never been successful and ended up replacing the carburetor. Messing with the needle and diaphragm has not always resulted Joy either. You can leak check the needle valve by plugging the inlet hose and pull the fuel out the primer bulb connection using the primer bulb. Cleaning the carburetor with the small holes and blind fuel passages can create a problem by plugging the tiny fuel feed holes you are trying to clean.. In the newer machine the adjustment screws are not removable preventing the cleaning of the needle clearances.
This is a bumpy road well traveled by many!!
Hope this has been a help?