Why will my scooter start hard after sitting for a week?

I have a Honda Lead AF20 from 1998 (I'm not sure, though) and in it 36200 km. I bought it at about 35900 km but we (I and my father) repaired or replaced a lot of things on it. They were: battery, piston, cylinder, cylinder head(not sure if it's called cylinder head but it's the thing in which the plug sits)self-starting electro-motor, bendix, bearings, seals, and there was something ugly amateurish work on the carburator with the seal of the(/a) needle going in the carburator. If I'm not wrong, that was the choke.
So, the problem is that after sitting for about a week it really struggles to start. The battery and the self starting is good, there is very good compression and the pug is also good. The self-starting electro-motor has to turn the engine for about 20-30 seconds to get the engine going. Whilst the motor struggles to start the engine, I can smell petrol and when it sstarts, it coughs out a lot of white smoke. It did it before and after the repairing, but when it sits for only 3-4-5 days it starts at the second I press the start button. If the waiting exceeds the one week, it will not get worse, I mean, it sat this winter for at least 2 months and after 30 seconds it started.
After the struggling starts it is weak until the white smoke comes out of it.

What can cause this problem? Might it be carburator or fuel-pump related?
Thanks for your answer:)

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