My Mum worked for a vegetable pre-pack plant who used shed loads of cling film which came with thick and long cardboard tubes so, knowing nothing, I went and wound a Tesla secondary on one. Then nothing happened.
https://www.youtube.com/watch?v=JI3VpKF8IF0
I'm not sure I understand it or perhaps the connections and reason for the spark gap. It appears to be two resonant circuits. Your Mains Transformer, with Leakage Inductance plus the Primary Capacitance and your Tesla Transformer with its Parasitic Capacitance and Leakage Inductance.
If you drive an LC tank at its Resonant Frequency the voltage in the tank, assuming the drive exceeds the losses, then the voltage across the tank builds up. Take a guess that if you manage to tune your Mains Transformer Leakage/Load Capacitance Resonant Frequency to your Tesla Transformer Leakage/Parasitic Capacitance Resonant Frequency you get a multiplication of the effect.
Take another guess that the nothing happens as these voltages build up until the spark gap breaks over. That will happen when the Load Capacitance hits peak voltage dumping serious current into your Tesla Transformers Primary and things go Sparky Sparky.
You need to rate your Primary Capacitor for the AC voltage it will see at resonance and at the point the spark gap breaks down and the resonant current it will see. Basically, someone else can tell me I am wrong, a DC, generally electrolytic, will not cut it even if you cunningly connect two of them in anti-series.
Then you need to find the right capacitor and you will also find no meaningful data unless you look for who bought up Siemens or Philips AN Other where they will give you lifetime, for applied voltage and frequency, and pulse rating graphs.
You also have to, once again my guess, match the Primary and Tesla Tank resonant frequencies to get a result or you could...
https://www.youtube.com/watch?v=wWIeUsnqkRk
Go mad like these Mad People and implement a half bridge LC resonant mode converter to remove your Primary Transformer and drive your Tesla Transformer directly. I know not a lot about these things but it seems more likely that you might want to consider an LLC resonant mode converter with, adjustable, primary side current limiting.
That is a thought or possible fantasy but if the Mad Men, unfair comment, got theirs to work, it looks like the feedback method used, and you are Hard Enough and use your Search Engine Fu skills to find out about LLC converters you might also believe there is a Ghost of a Chance.