So I thought to the varicap (varactor) which can be voltage-controlled by a micro (Arduino in my case) but these kind of components have a so low capacitance values (in the order of few pF) that this makes them useless for my purposes. Furthermore, in the presence of so high voltage peaks, they will be destroyed in microseconds!
Maybe what you need is a vacuum variable capacitor. These are usually driven manually, or by a small motor/gearbox.
They are available up to several hundred pF and 10s of kV rating. Have a look here:
https://www.ebay.co.uk/sch/i.html?_nkw=variable+vacuum+capacitor
JimB