Transformers confusion with output voltage and number of iron cores used
Why do I observe an increase in output voltage when I add more iron
  cores to my transformer? How do I explain this using the laws of
  physics?


Think about it it must be better than air, why would this happen? One is because there is less leakage of magnetic fields. The other is that magnetic fields flow better through iron than air. 


  Is there a relationship between the number of iron cores used, and the
  output voltage in the transformer?


You would really have to describe what the total construction of the transformer looks like. Since this is an academic exercise I'll give you a hint:


Source: https://en.wikipedia.org/wiki/Magnetic_core

Also: 
Air is less efficient, iron is more efficient. The more iron you have the better magnetic permeability you have between the primary and secondary. You could think of air as more of a magnetic resistor than iron. The more Iron you add the better the magnetic permeability you have between primary and secondary cores. 

In short: 

to find the magnetic field through the core use this.

\$ V_p = N_p\frac{d\Phi}{dt}\$

Then find the B field:

\$ B=\frac{\Phi}{A} \$

Then insert the magnetic permeability of the material:


Source: Slide 34 https://slideplayer.com/slide/6877632/
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