On most Hondas (and some other brands) the choke is controlled by a thermowax choke actuator. As the mower heats up, the wax in the actuator expands pushing out a pin, which in turn controls the choke..
It's a tiny little thing that looks like this (and costs ~$10).
When they go bad the mower will start fine, but stop once it gets warmed up. It will also be difficult to get the mower started again until it has cooled down.
It's a known problem on some models of Honda HRX mowers... with the simple fix of removing the choke, pulling out the thermowax actuator and replacing it with a new one (~10 minutes of work).