Hi Liam.
The core in your transformer picture has rounded corners, a style I'm used to seeing in tape-wound laminated steel cores.
To follow Werner Engel's question, are you sure it's ferrite? The magnetic properties and the efficient frequency range are very different for steel.
Here is one of many reviews of core material categories.
http://www.ti.com/lit/ml/slup124/slup124.pdf
Either way: since you can take the core apart, I suggest that you put it together
without the high voltage windings, while you use it in experiments for learning.
You can make your own experimental windings on folded cardboard bobbins.
The electrical behavior of a primary winding
with an unloaded (open circuit) secondary
is practically the same as that with _no_ secondary.
All models are wrong; some models are useful. -- George Box