You have the standard tamper-proof pilot screw, so you shouldn't plan on adjusting it, or removing it for cleaning. That isn't too serious, you can clean around it if cleaning the whole carburetor becomes necessary. In real extremity I can even tell you how to adjust it.

I suggest you see if you can put a small container underneath that retaining screw that is in the center of the bottom of the float bowl. If you can put a container there, turn off the fuel and loosen the float bowl retaining screw. Fuel (and just possibly water) will flow out into the container until the float bowl is empty. Remove the container and look closely to see if it contains any water as well as the fuel. If it does, you've found your problem. Whether there is water or not, take out the retaining screw. The float bowl will drop off into your hand. Check inside the bowl for signs of dirt or water. Put the container back under the carburetor and turn on the fuel tap. See if fuel flows full stream out of the entry port on the underside of the carburetor body, above the pivoted side of the plastic float.

There is a very good chance you have water in your float bowl - it fits the circumstances well, since the problem happened suddenly when the machine was in the hands of an unreliable amateur. However if there is no water, and you haven't found a problem when you get to this point (either water or restricted flow), it is time to remove the main jet. Here is an internet check sheet for servicing a Honda carburetor approximately the same as yours (the sheet is actually for a GX or GXV carburetor, so it is somewhat more advanced than yours, but has the same functions and layout):
[Linked Image]

You can see that the main jet has a thread on the outside, and it holds the emulsifier (which they call the "main nozzle") in position. If you have a suitable stumpy screwdriver which will fit below the carburetor you may be able to insert it vertically into the bottom of the carburetor, via the thread that the float bowl retaining screw came out of. At the top of that thread is the main jet, which has a screwdriver slot in its underside. If your screwdriver is a good fit in that slot, unscrew the main jet so it drops out. The emulsifier may drop out after it, but it probably won't. Inspect the hole in the center of the main jet and see if it is perfectly clean. If it is not completely clean, you will have to remove the carburetor and clean it in detail. That will probably destroy at least one of the gaskets that sandwich the plastic insulator between the carburetor and the intake port on the cylinder head. That is not an entirely bad thing, since if anyone has ever messed about with the carburetor retaining screws, they have very likely split one of the gaskets anyway, and your symptoms are a very good fit for a split and leaking insulator gasket.

I suggest you follow these notes this far and report back on what you find, whether everything is now fixed or not. If you haven't found the problem, the next step will be to remove the carburetor and go through a standard cleaning procedure, which I can help you with in detail. If that turns out to be necessary you will end up with a clean carburetor and new insulator gaskets, which should eliminate just about all of the potential causes of your symptoms except internal mechanical problems such as a leaky valve or incorrect camshaft timing.