Transformer question regarding efficiency
Without a diagram, I can only assume that 'adding cores' to a transformer means increasing the effective area of iron within it. This interpretation is also consistent with the behaviour you describe.

Let's assume it's some form of teaching transformer like this (image from betterequipped.co.uk)



where you alter the iron area core you introduce into the magnetic circuit. 

You can start with no iron, then 'add cores' by putting a few thin laminations into the magnetic loop, then go on to increase the area of iron to fill the entire 'iron area' with iron.

You need to make a distinction between magnetic field (which is a per-area measure) and magnetic flux (which is a total measure within the centre of each core).

As you add more iron area, you increase the magnetic flux passing through the cores, and it's the flux, not the field, that controls how much voltage is generated per turn. More flux, more volts.