My chua 50cc scooter (only 6 weeks old) will only start with the kickstart. why can't I now start with the electric pushbutton?

If you buy the pinion you also have to buy the rest. Also on the axle is a circlip wich you must use. Otherwise the Carter where the axe is going in will be worn out. That is a problem if the starter is not working. They use the kickstarter to much. Its only a help not a thing to use each day. The parts for this are here.
Its number 4 on the drawing you need.
https://goparts.eu/en/blueprint/peugeot/scooter/2011/fight-3/clutch

And here is the number 3 thats most of the times is also worm out.
For using it properly you need Numbers
1-3-4-5 and number 6 is the ond you Jane bought already.
https://goparts.eu/en/blueprint/peugeot/scooter/2011/fight-3/drive-shafts-kickstarter

tighten up the bolt in the lever next to the splined shaft.

Look on the engine most to all china made scooter's have these engine's (136qmb)(gy6) and other variants of these motor's for wire's they follow the frame once you take a good look where every thing goes I recommend finding the CDI it's a little box with about 6-8 wire's plugged into it this is the brain of you're scooter if you want part's just search (50cc chinese scooter parts) and most if not all the part's will fit you're motor.

try a new battery

If it is just the kick arm, that is simple. If the shaft that it connects to is bent, this is very difficult to access. Please have your Honda Shop take a look at it. If the electric works, you might want to consider using it that way. If there is internal damage, it might not be worth repairing this 10 year old scooter.

the kick stater machanism has a spring on it
which may be brocken
Cheers Rob

probally under the seat