So that's a first-gen HRX, with a standard manual choke.
Step 1. Confirm the choke plate is fully opening and closing. Does the engine run/start smoother with the choke closed?
Step 2. Consider replacing the carb (full list price is only $16.23).
Step 3. Triple check the position and orientation of each gasket, spacer, insulator, etc. It is VERY easy to get one on upside down or backwards, and in many cases, the carb won't be able to vent.
Robert,
Thanks for the list of things to check.
1) The choke plate is fully opening and closing as the throttle lever enters/leaves that portion of the control
2) I did replace the carburetor with a brand new one from a Honda dealer. That did not result in any change
3) I checked positions and orientation of gaskets and such, confirming they are installed correctly. This problem started happening prior to having anything apart so I didn't think it was anything related to gaskets.
I drained the tank, added fresh gas, and still not really running. Still the coughing, sputtering, "idling" even with throttle wide open, etc. I couldn't get it to run regardless of choke, full throttle, idle. One time, It did start and run almost perfectly for about 15 seconds and then died. Wouldn't restart.
Is there anything I should be checking regarding the ignition? Could there be an issue there?