And thats the twin carb Zenith model? Sounds like fuel accelerator pumps on the carbs to me, the levered and rubber booted piston looking devices on the side of the carbs. A very common problem with Zeniths thats easy to fix with replacement parts. Otherwise can be distributor advance mechanism, from ignition perspective. Or valve clearance adjust.

Regarding white smoke, dunno, and wouldnt worry about it if the car doesnt burn alot of oil. Meanwhile check the oil while the engine is idling - blown headgasket will show water bubbles on the dipstick.