As others said, the carb is the problem. There is a small passage plugged. This is a common problem with Honda gensets if stored with untreated gas, and with many small engines as well.
Run Seafoam in the gas. Start it, run it with the choke for a little bit. Let it sit 24 hours. Repeat every day for several days. See if it cleans up. I've had it work! Usually you'll see progress every day. Another good chemical for cleaning up dirty carbs is Sta-Bil Marine or Sta-Bil Ethanol formulas. Use the clean up ratio, which is twice the storage ratio. I've had the Sta-Bil work when the Seafoam failed. But don't use the old original version of Sta-Bil; it's worthless with today's gas IMO.
Then use the Sta-Bil Marine or Ethanol in gas, at the storage ratio, in the Honda from now on, or at least when the gennie is going to be sitting.