With each separate voltage multiplying circuit, you have to have a "reference" or "earth" ot 0v point for the high voltage to
"conduct to." If you allow a corona discharge, it is using the ground as an earth point.
With the circuit you have provided, one arrangement is producing a positive potential and the other half is providing a negative potential and there is a varying voltage between the two due to the supply voltage fluctuating. The capacitor effectively joins two two halves and provides you with a voltage that can be considered as 0v from the negative terminal and twice the potential from the positive end.
But this means the potential at the power-in will be half the high potential you have generated and this must be considered.