This article pertains to the Honda auto-choke.
The little box on the cyl head houses a bi-metal directional valve that redirects engine vacuum when it (warms up) de-activating the enrichening circuit in the carb. If you think that the hoses are marginal, change them first to eliminate that (cheap fix). If it still does it, take the little cover off and try not to ruin the gasket. Take the valve unit off of the head and spray it out real well to make sure oily gunk is has not built up and disturbed the lttle seat/plug on the thermal arrm. Don't bend the arm, it's calibrated, but lift it gently to spray it out. When you go back together, the cover has to seal properly or you'll lose vacuum so use a very light film of silicone if needed.
If all that doesn't work, the problem has to be in the carb itself, but it's usually not.
The little, tiny, microscopic, jet hole - that can hardly be cleaned out properly because of the akward drill angle isn't clogged in the bottom of the bowl. You should be able to spray contact cleaner or something with a small tube (WD), holding it tightly, and get a decent flow through it. These ports are so small and in the worst possible location feeding from the bottom of dirty bowls that they inevitably clog and cause fuel flow problems with the auto choke system.
Vacuum shouldn't be the issue if the hoses are new and it's coming off of the intake port, it should have enough vacuum to draw the fuel from the bowl port/jet. If the bystarter foam filter is clear, it should be able to draw enough air too. Can't replace that foam with "closed cell" sponge material. If the bi-metal control box housing is sealed it should work. If the plastic housing is not cracked it should hold a vacuum.
It's really not that complicated. Just think about it as a simple little fixed-venturi carburator with no slide control. When cold, it supplies a pre-determined / controlled mix of air (bistarter) & fuel (float bowl). It slowly transistions to close off and seal the intake vacuum port as it comes up to running temperature where the carb takes over.