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.