In particular, why would the engine start to run roughly only half way through the circuit, and not on initial application of full power with carburettor heat on?
If you have carburetor ice, you may have little or no response during initial application of carb heat. Roughness develops as the ice melts.
If the mixture is rich already and a low compression engine is doing little work to scavenge an unburned mixture, and the plugs are low tension or weak or the gap is wide or there's little heat in the cylinders, or there's an oil fouling issue with leaky valve guides (and so on and so on), fouling can be an issue with one or more fouled plugs. Just one or two cylinders fouled can really shake an engine.
If the carb air temperature was outside the icing range when carb heat was applied, the carb air temperature can be moved into the icing range...and carb heat can cause ice to form. Lacking a carb air temperature gauge, you're taking a shot in the dark by applying carb heat. Generally the manufacturer tries to err somewhat on the side of caution by making it too warm rather than not warm enough or in the icing range...but on a cool, humid morning, you can cause ice under some circumstances by applying carb heat.
If ice has been forming and the throttle is opened, the engine may just quit. It may be running at a lower power setting, but you can see a variety of potential things occuring if an occlusion exists adjacent to or on the throttle plate, or in the venturi. Opening the throttle is throwing a lot if air into that engine which may not be matched by the fuel flow; you may just put the engine out, send a slug if ice down the carb, or have other problems occur.
Dark smoke is indicative of an excessively rich mixture, or a mechanical issue such as a hole in a piston or a failed valve guide.