It didn't used to be a problem with Hondas, but with the ethanol fuel or old fuel, the jets are getting gelled up. Your symptoms indicate a probably clogged jet(s). Here are some pictures to show what I mean.

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Buying a Honda carb from Honda of Plano, TX is so cheap, it doesn't pay to save a badly damaged carb anymore.

I had not seen varnished Honda carbs until recently. They seem to have escaped this dilemma. Here is what you're looking for.

Most likely, you'll have to remove the carb. Make careful note of the linkage and how it comes off. Remove the bowl nut and in the center of the carb you'll see an opening just under the bowl nut. You'll need a special, thin screwdriver to remove the first jet. It has a slotted head with a center jet. Above that is the long tube with many small holes. Both pieces have to be thoroughly clean. Use carb cleaner to clean the rest of the carb; any orifices. Do not overtighten the jet when you put it back in. Very light torque; it isn't going anywhere. This should solve your problem. This carb worked like a champ when I was done. Good luck.

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If you're going to buy a new Honda carb, my first choice would be the Honda dealer. The carbs are incredibly cheap - no S&H from the Honda Dealer.