Q:

Transformer question regarding efficiency

Due to the question being discussed in length on the comment section, I would like to address my question further with a new post linking from the discussion here: Transformers confusion with output voltage and number of iron cores used
So with the help from this community, I understand that more iron cores means a greater magnetic permeability and less magnetic leakage.  But does this mean that the magnetic field increases?  
And assuming this is true, if we have a transformer with many iron cores in the center as compared to air, then the one with more iron cores will increase the current in the primary coils.  So what I am failing to see here is, why is a greater magnetic field established (due to more iron cores used) increase primary current of the transformer?  
I know there there is a magnetic field established when there is a current carrying wire, but I do not understand the relationship between magnetic field and current established.  that is why does a stronger magnetic field established increase current in the primary coils is my key source of confusion at the moment with transformers.  
Eventually I want to try to understand why more iron cores used increase the out voltage value of a transformer(secondary voltage).  This is my understanding at the moment with many 
$$\text{more iron cores}\rightarrow \text{stronger current in primary coils}\rightarrow \text{less current in secodary coils}\rightarrow\text{stronger secondary voltage }$$
I am not sure if this pathway to my understanding is correct, but I am stuck on the first arrow from more iron cores means stronger current in primary coils. 
I would appreciate the help and clarification.   
EDIT: This is somewhat what the experiment the professor was doing in front of the class today.  

A:

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.