Advice for painting kitchen cupboards

[Chanel05]

I've been toying with a new kitchen for 4 years but now that I have a baby due in a couple of months, it definitely isn't happening!

I looked into painting my kitchen cupboards, looks fairly easy in the prep and pruning etc. However, my kitchen is a very cheap model and it has this plastic coating over the top. Would I need to remove this and paint underneath? Or would I just sand, clean, prime and paint the coating? This is the only cupboard where it's coming off.

I used this in white, didn’t even use the whole tin.
www.johnstonespaint.com/products/cupboard_paint
It goes on best with a foam roller and doing a couple of coats really thinly was better than trying to do one thicker coat.
We were undecided if we should strip off the coating and paint bare mdf or paint the coating. As we had removed a cupboard we experimented by half stripping the coating and trying a bit of paint on each surface.
We decided on leaving the coating on, it just had a better feel. However, 7 months on there are a couple of doors and a drawer that now have chips of paint scratched/come off so I think stripping would probably have been better.

Thanks :). How would it be stripped? I've looked over it this morning and it looks firmly stuck on!

Tbh I have no clue. The door we no longer needed was already peeling through heat (it was too close to the hob) so it was already lifted. Maybe heat is the key?
A quick google suggests things but you would have to know what your coating is maybe?
www.dianellapolishing.com.au/blog/kitchen-cabinets-how-to-remove-peeling-vinyl/

m.youtube.com/watch?v=EfNuB1qi9WE

www.home-dzine.co.za/kitchen/kitchen-remove-foil.html