Thank you for your advice "one poster". The bathroom floor under the toilet is near perfectly level. The plumber mentioned that during the installation and we checked it together.
I have attached a photo of inside the cistern which shows the water level at the 6 litres mark, the max the mechanism can be set to. I then placed my hand over the central part of the mechanism, namely the upper part of the red pipe which would remove overflow water, and effectively blocked the overflow. I was then able to add an extra 3 litres before reaching the holes at the back of the cistern (for securing to the back wall, but not used in my case).
A flush with the extra the extra 3 litres was very much improved. It cleared the bowl perfectly as it should be able to do in one flush. I feel that a 9 (instead of 6) litre flush into the same 'controlled flush bowl' will solve my problem. I hope that I'll find a mechanism that can fit inside the cistern and allow it to fill to 9 litres, which is quite near the top of the cistern.