My Friend, now the true picture comes about. I apologize, I did not realize the picture of the Lawn-boy was the actual one in question as current Lawn-Boy mowers do have Honda engines on select models.
What you have is a Lawn-Boy Engine with a Honda Recoil Starter. It is most likely smoking because the muffler, which is located under the deck, needs to be separated and cleaned. It is all carbonned up. Common when junk 2-cycle oil is used. Clean it, messy but simple. Use premium fuel and Lawn-Boy oil. It will run almost forever.