It IS that auto choke if it is a Briggs (I know you mention it's a Honda, but many of these mower brands will use several motor lines on one platform, not sure if Honda does). I had that problem on my newer Snapper and the autochoke was replaced under warranty. But even the replacement doesn't work right. So when I first start it, it runs, then instantly starts sputtering. So now I have the air cleaner off and as I start it, once it starts, I open the choke plate until the engine warms enough to hold it open on it's own.
The mechanism uses a similar theory mechanism that old carb engines with chokes used. It has that bi-metal coil that reacts to heat and as it warms it stretches, actuating the linkage to the choke plate, opening it.
How it works on the Briggs, is that coil is mounted behind the muffler, then has a linkage that routes over the top of the motor, to the choke plate. And once it starts and warms, it should automatically do it's thing. So to test it out, take the air filter housing off, start it, and then use a screwdriver to hold the choke plate open. If once it warms and you remove the screwdriver, like mine, it'll run fine. If not, you know this is the issue.