That thing that looks like a fuel solenoid is actually a carb heater. To assist warm up time and to help prevent carb icing. Something to check other than making sure that all of the carb jets are clean is to check the water in the battery, and also check the output to the battery. Voltage regulators will overcharge causing it to boil the battery dry. the dry battery issue prevents the battery from accepting the charge and will try to dissipate the voltage causing erratic run issues. Something else to check is the valve clearances because over time the valve and valve seats wear closing up the gaps causing it to float the valves.
If all that checks out, remove the variable slide venturi from the carb. take out the long needle hanging down from inside of the slide and move the E clip down one notch toward the narrow end to richen it up a little and see what happens. And if the spark plug hasn't been replaced in awhile I would go ahead and replace it. I have seen spark plugs on Honda ATV's do weird things.