The purpose of the emulsion tube is to aerate and mix the fuel that passes through the main jet heading up to the venturi..
In usual circumstances the emulsion tube admits air from some of the drillings and adds some fuel from others....the fuel passages are extremely fine and difficult to see - it's a blast of compressed air that's needed to sort that, with luck.
The carburation is in three stages - at least on a more complex carburettor system.
1) The idle stage, where most of the fuel passes through the idle jet, and aerated with an air bleed passage;
2) The intermediate stage where the emulsion tube helps to deliver fuel into the venturi when not on full throttle;
3) The full throttle stage where the idle jet and emulsion tube are largely redundant.
However, who can say of the accuracy of this when applied to simple carbs by Honda or Briggs and Stratton are anyone's guess!
Anyway Rich - to get your carb performing again the emulsion tube and idle jet do need to have to come out - the chances are the alloy corrosion, which will surely be there to some degree is interfering with things.
Pattern carbs are available at a fraction of the Honda price, of course.
John