It could be the carby, a kit to replace the diaphragm etc would not go astray, I have had a number of these engines apart and it is almost always the carby, the diaphragms get stiff over time and do not operate as they are designed to do. I read somewhere that it is advisable to do this every 12 months, an overkill but does indicate that it is important, the diaphram is the pump that gets the fuel up to the engine.
Regards Beech