If you can 'fix' it temporarily by choking it, the problem is fuel related.
I'm willing to bet that even though you think you've cleaned the carb, you haven't. Honda carbs are prone to gumming up if left standing, and nothing short of an acid dip followed by an ultrasonic clean will shift the crud that has accumulated. I've had to replace carburettors in the past due to internal corrosion.
It's worth having another go at cleaning. Use a piece of wire stripped out of some electrical flex to poke the jet through. 3 strands twisted togeather as tight as possible should JUST go through.
phil_saunders@bigfoot.com
Phil Saunders, September 2008