Hello and Welcome to Just Answer,
The problem sounds like you have a plugged up passage in the carburetor in the starting circuit. When you use the electric start it cranks it over faster so it lets it start. Once it is warmed up then it is affecting the starting even moreso. I would rebuild the entire carburetor using a complete carburetor rebuild kit. I would also replace the fuel lines as well.
I forgot tomention, I replaced the carburetor last year - put in a new one. Prior, the engine wouldn't even run for more than a minute before stalling and if it did run longer, it wold stall under a load like blowing the snow. the new carb made it better but still not optimal running. As for the gas lines, gas pours our pretty fast. A carb kit a relatively inexpensive and perhaps bad gasoline has already fouled up the new carb. I also just bought a new spark plug at the store and will install that. In my opinion, an small 4-stroke engine should start in one or two pulls. This one has never fallen into that category, unfortunately. Any further suggestions wold be greastly appreciated.
Kind regards XXXXX XXXXX new year.