If you have good spark, and the engine internals are (presumed) good (piston/rings/valves/ etc.) then poor performance is usually a sign of incorrect levels of air/fuel being drawn into the engine. This can be anything from a clogged air filter, vacuum leak, partially blocked carburetor (especially the main jet), incorrectly operating choke, and so on.
Any other symptoms (consistent white or black exhaust smoke, excessive vibration, hard effort on the recoil starter, backfiring)?
So, air filter, clean and no damage?
Pull the air filters off and operate the choke. Confirm it closes all the way and opens properly?
When you cleaned the carb, did you use aerosol carb cleaner
and compressed air? Hard to do the job 100% without both of these. Did you fully remove the main jet and nozzle? They both have extremely small holes that are easily clogged and overlooked during cleaning.
When you cleaned the carb, did you replace the gaskets and get them and the spacer/insulator all back on correctly? A "backwards" gasket can cause poor performance.
Finally, if all the above check out, it may be the internals of the carb have some gunk/slime that may be the culprit. Often, it can be faster/easier to simply replace the carb with a new one, as it can be time consuming to otherwise troubleshoot. For your older HR-series mower, list price on a real Honda carb is $45~$60 price, but you'd probably pay a shop more if they professionally cleaned (ultrasonic) your old carburetor.
Here's a link to proper cleaning of a Honda carburetor, FYI:
How to Clean a Honda Carburetor
