There are at least two ways you can get air flowing back into the fuel hose from the carburetor. One is if there is a blockage in the fuel hose or fuel tank vent: when a vacuum forms upstream of the fuel pump, it doesn't perform well as a vacuum pump and there will be an air leakage into the pump, one way or another. The other way is a defect in the carburetor that lets air in - such as a perforated diaphragm or a leaky gasket. And yes, the diaphragms can only work if they are air-tight.
I suggest you get a kit for the carburetor, which will include gaskets and diaphragms. Then follow the process in the manual as far as possible (you may not be able to access some special tools), cleaning, clearing, adjusting and putting it back together with the wearing parts replaced. I wouldn't recommend that approach with a bigger carburetor for a B&S or similar engine - I'd normally just find and fix the problem - but those little carburetors are fairly complicated and rather sensitive. A blind, largely mindless overhaul seems better than having to acquire watchmaking/forensic skills that might be fairly useless afterward. Of course if I weren't a bit on the heavy-handed side myself I might see this differently. If fixing watches is your game, working through a diagnostic process might work. However it seems to be cheaper to buy a kit than to chase individual bits as you belatedly find defects in them.
