The "choke" on the Honda Rancher is *not* a real choke (which reduces air intake), it is an enrichment circuit (adds fuel).
Your symptoms mean not enough fuel is getting through the carb.
1. Fuel supply- check tank, petcock, filter, floats, primary jet, fuel pickup tube (not sure if this carb has one, but likely). You can rule out and correct most of these by pulling the carb bowl and then turning the petcock to on/pri position.
The pickup tube would be a longish brass tube that sits in a small tube in the bowl. There should be a small hole in the bottom of the pickup tube area in the bowl.
2. Too much air- MUST run it with air filter on and sealed (and oiled- maybe it dried out?), and make sure the intake tube is well sealed to the airbox and the carb...and the carb to the cylinder. The airbox must be sealed (did the foam seal rot away?)
3. Wildcard- spark plugs are known to get intermittent and die on these ATVs (we have a rancher with the electric shift). Spend <$5 and get a new one to be sure
Running Seafoam through the carb can help break down the varnish (I'm agreeing with the bad gas advice given by others). The results may be a sludge that either gets run through the cylinder and out, or it could clog up the jets. But they're already clogged, so hopefully it will help.
Hope this helps.
- JC