Are you 100 percent sure the fuel in the can is gasoline- not diesel or water for that matter? If not, I guess I would start by dropping the bowl of the carb to look for water /debris- cleaning the whole fuel system if need be.

If your mower has a polymer rubber primer bulb, and you are using this to start the mower, successfully, then you are running the engine on the primer, and it dies when the primer fuel runs out. You can keep the engine going by continuing to prime it every few seconds, but that is inconvenient to say the least. Our mower had a similar problem this spring. The solution was to add carb cleaner to the gas, and run the engine. If you can get the engine started, you can spray carb cleaner into the carb, and keep it running that way. If this doesn't work, the carb will have to be taken apart, cleaned piece by piece, and re-assembled.

Sounds like a stuck needle and seat. My guess is the bowl is not filling with fuel. I would run carb cleaner through as well as a cap of MMO or good fuel conditioner/cleaner in a few tanks of fuel if you get it running. If this does not work remove the carb and give it a good cleaning/rebuild. You may get away with just blowing compressed air through it as well as cleaning the needle and seat assembly. If it were me I would dismantle it since it will be off.