Okay, here's what our tech guys say:
1. When you tried to remove and clean the main jet, spraying air/cleaner into it may have impacted the possible debris/gunk into the emulsion tube. See part #14 on the attached drawing. So, that main jet and emulsion tube should get removed and cleaned, along with a complete cleaning of the entire carb. As a new carb is $192 list (yikes), you may wish to have a Honda engine dealer do an inspection first, then it's on them to get the engine running correctly.
2. FYI, the fuel cut solenoid works opposite what you'd think on other engines:
First, it get a steady +12V from the genset when the engine is running, but the ground is not connected, and the unenergized position of the plunger on the solenoid is retracted.
Next, when you turn the engine OFF, a ground (-12V) is sent to the solenoid, causing it to extend and block the fuel from flowing through the main jet. It stays in this position until the genset stops supplying +12V to the solenoid, then the plunger retracts.
So, it's less of a "fuel cut solenoid" and more of a "backfire prevention device." If the main jet were to continue getting fuel during engine shutdown, it's likely there would be a backfire through the muffler.
Hope this helps, Mike. Here's a link to find the nearest Honda Engine Dealer:
Honda Engine Dealer Locater
Here's the image of the carb from the parts catalog (attached).
Caveat: I work for Honda, but the preceding is my opinion alone.