Time for a carb strip down I'm afraid.....not quite as daunting as you may think, but a can of aerosol carb cleaner is a must, and I don't do these without a compressed air supply.
Clean all of the external crap from the carb, and then remove it. The fuel pipes need to be a tight fit.
You'll see that the ends are squarish, one end may have a central screw and the other two or four screws.
Under these square ends lives a rubber diaphragm. One is called a metering diaphragm (usually under the single screw end) and the other is a pump diaphragm. This one pumps fuel from the tank to the carb and then back again - as there is no float chamber to act as a fuel reservoir.
Basically you need to remove every movable piece from the carb and flood the thing with carb cleaner, through all of the drillings. This also involves removing the needle valve which has a pin held by a single screw.
Beware the tiny spring in there!
At this point I'd be blowing compressed air down each and every drilling.
Looking at the diaphragms, if they are split or damaged, they need to be replaced.....if the carb is a Zama make it will have a C1Q number to help get spares. Never touch any part with kitchen paper or similar.
Careful reassembly, and turning each mixture screw out one turn should get you some result!
Good luck if you decide to get stuck in.
John