The most common reason why your push mower will only stay running with spraying gasoline into the carburetor is a fuel delivery issue.
First check to make sure that gasoline is actually reaching the bowl on the carburetor.
You can take the bowl off the bottom of the carburetor and see if it has gasoline in it.
If the bowl has no gasoline in the carburetor bowl then you either have a clogged fuel filter, clogged fuel line or even something blocking the outlet of the fuel tank.
The needle for the float in the carburetor may also be blocked or not opening up.
If there is gasoline in the carburetor bowl and the push mower engine will only run when you spray gasoline or starter fluid in the carburetor then you most likely have a clogged main jet or the bolt that secures the bowl to the carburetor may also be clogged.
A carburetor bowl bolt that is the center bolt should have around 3 small holes that allow the fuel from the carburetor bowl to enter the carburetor to send through the main jet.
The holes in the carburetor bowl bolt can sometimes get clogged with varnish from old gas or even something else.
Try using a small piece of wire to clean the holes in the carburetor bowl bolt and that can sometimes fix the issue.
Also you may need to remove the carburetor and soak the carburetor in some carburetor parts cleaner and then blow the main jet and other ports of the carburetor out.
You can also replace the push mower carburetor for pretty cheap as well by buying a new carburetor online through eBay.
I had an issue with my push mower as well only running when spraying gasoline into the carburetor and I even cleaned the carburetor by soaking it.
I then forgot about the carburetor bolt with the holes that hold the carburetor bowl on and after cleaning it with some wire and dislodging a piece of debris the push mower is now running good.