Arcs like to start where there is a so-called corona. A corona is like a one-sided arc: it emanates from a single place - and this will be from a point of high voltage, particularly from a sharp point. This is why Van Den Graaff generators use spherical poles: to reduce corona and risk of arcing. So: reduce sharp edges everywhere but on the spark gap.
This will have a second benefit: a corona leaks current. So reducing sharp edges everywhere but the spark gap prevents unwanted current loss and so reserves more for the arc where you want it.
What size arc do you want? A very rough guide is 1kV required per mm gap. If you really are generating 10kV then I'd be very surprised if you don't get a spark over a few mm. It might also be helpful to measure what current output you get when you short the output of the coil with an oscilloscope in parallel with a small resistor, so acting as an ammeter. The higher the current it can output, the greater the spark. (And the more dangerous!)
Also remember that at very high voltages, what would normally be considered an insulator actually becomes a relatively good conductor. Insulation with 10MOhm resistance across 10kV will be carrying 1mA! (Meaning 10 watts disappears into your insulation. To reduce this, try operating in dryer air, use the best insulators you can, ensure greatest path between conductors, and ensure all paths are clean, eg to stop grime conducting.
Also, consider unwanted capacitances: at high frequencies (including sharp pulses) a small capacitance between conductors can absorb AC power.
Also, consider that many components are not ideal. Your capacitor will leak. Any stored voltage will leak DC current between the plates. Choice of capacitor will be crucial; check its data's heat for specs. Your diagram says the cap never exceeds 0.02uF - this is non sensical since a capacitor's value is fixed! 10nF capacitors will always be 10nF, nothing else.
How are you measuring current in your circuit? Why do you have the toggle switch? I bet this is arcing since I have rarely seen toggle switches that are designed for more than mains voltage! Very dangerous too, I wouldn't put my finger anywhere near that. Where did you get your circuit from? I'm also wondering if there should be a diode somewhere.
If you want more help on this in the future, more details would help: pictures, specifics of your layout - at such high voltages, such things are crucial, unlike for low voltage circuits.