Hi,
It's hard to make out what you have going on by looking at your photos but I'll take a shot at it.
The coil spring to the right of the plunger and bellows is supposed to fit over the "golden" part so that it is turning with the throttle plate shaft as the carb linkage is moved on the opposite side of the carb.
But...it appears....that something is missing. There should be a black plastic arm on the part with the coil spring that has a sort of little bowl at the end. The bowl "snaps" over the ball end on the plunger pin. Maybe it's there but I can't make it out.
If it isn't there, then someone has made the somewhat classic and unfortunate mistake of being too rough with it and snapping it off of the main piece. If that's the case, then the entire carburetor would need to be replaced IF you decide that you MUST have the accelerator pump working as designed as that part is unavailable via the Honda parts system. It's considered an "integrat" part of the carb.
However, the reason I capitalized IF and MUST in the paragraph above is because you don't really need to have THAT accelerator pump operational to continue using the carburetor.
I have worked on hundreds or, possibly thousands, of carburetors with accelerator pumps and this particular one pretty much strikes me as being next to worthless. I have literally removed the piston and spring from these because of binding up and no parts were immediately available and the boat operators (rental/dock hands) never noticed the difference. Not because they were dense but because that pump just doesn't really do much.
One reason it is so ineffective is that the piston to bore tolerance gets very big early on due to metal to metal contact and no seal. Gasoline is not a lubricant either. At any rate, the check ball that is supposed to seal the inlet passage at the bottom of the bore gets loose early on in the carburetor's life from vibration and doesn't seal well, if at all. So, even if the piston had a seal, it would likely push as much or more gas BACK through the feed passage and into the float chamber as it would push gas out the accelerator delivery tube and orifice.
In other words....the accelerator pumps on these finicky but otherwise great little carbs is a poorly concieved add on to the older versions of this carb.
Bottom line? IF you don't want to buy a new carb, you don't really need to. Go ahead and use that one. Just make dang sure that the accelerator pump DELIVERY passage and tube are clear and that the orifice in the tube is not clogged. If these items aren't clear, the engine won't run correctly. I suspect that it's because they act as a sort of pressure vent for the float chamber although I don't know that for sure.
Good luck.