If the carb flange gasket is breaking up then that might allow air to leak into the inlet tract on the hot side of the carburettor, which would give lean fuelling.
Also, as other posters are suggesting, a compression check would be worthwhile.
Can you still feel good resistance at the kick start?
40,000 kms is a good run for British type air cooled 350. A top end strip down for decarbonisation, piston and valve checks and assesment of big end bush condition would be on schedule for that amount of engine run time.