Okay, that's an early production HRX217VKA...
First up, Honda issued a Service Bulletin back in 2008 for hard starting problems with a limited number of these mowers. Your serial number is in the affected range that might have a problem with hard starting. The fix is a redesigned/updated carburetor, which has an updated choke design. This problem may happen more often with fuel that is several weeks older and / or when trying to use the mower in cooler weather. Now I can't say for 100% sure this is your problem, but it would be a good place to start.
So, in your original post, you said you replaced the carburetor. Do you have the part number or package it came in? Did you get it from a Honda dealer?
The only correct carburetor for this model is Honda Part Number
16100-Z0Y-871. No other part number should be used. This carburetor is about $50 list price.
You can visually check to see by looking at the bottom of the carburetor. The
correct one will only have single center bolt. Older, original carburetors will have 2 bolts, one center and one drain bolt on the side:
The Service Bulletin advises to apply some HondaBond HT silicone to the autochoke heatsink when installing the new carburetor to ensure proper heat transfer and operation. When you replaced the carburetor, did you apply any type of silicone to the heatsink? How old is the fuel in the tank? Did you drain it all out and fill with fresh fuel? Are you 100% sure you got all the gaskets, insulators and parts installed correctly when you installed the new carb?
If you'll drop me a private message with your email, I can send you a copy of the bulletin as an FYI.
Finally, if you
do have the correct carb installed, and you
do have a good physical connection to the heatsink, it is likely there is another problem
not related to the Service Bulletin, but let's figure that part out first.