Without even one single word on what the desired effect of that capacitor would be, any suggestions will be guesses. There is some value that will tune the transformer secondary to resonance, and there is some lower value that will serve to reduce the ringing of the square waves edges a bit.
And there is no certainty that the circuit will function at all, given the unknown supply voltage.

 

A CD4047 can't source or sink much current, so the gate capacitances of the FETs will charge/discharge relatively slowly. That could result in significant heating.

 

If I make the big assumption that the circuit you show is known to work properly, and you're basically asking what capacitor should you use, then I believe you're after a type X, I'm not sure if a type Y would be appropriate here. Here is some reading on capacitors for high voltage AC like you have.

 

Should'nt it be something like 2.2 uF or 3.3 uF @ 440 V AC?

 

Looks like an inverter, just guessing but it’s working off battery perhaps 12v. It’s for 50Hz output.

The formula is I=C(dV/dt) it will depend on your load. You’re trying to filter the switching to a pseudo sine wave.

you may get better results with an inductor before the cap, never mind the transformer is performing this function.

 

If I make the big assumption that the circuit you show is known to work properly, and you're basically asking what capacitor should you use, then I believe you're after a type X, I'm not sure if a type Y would be appropriate here. Here is some reading on capacitors for high voltage AC like you have.

I would not make that presumption that the circuit shown in the posting is known to work properly. The resistor for the LED does suggest 12 volts supply, but since we are not given any hint as to what effect that capacitor is intended to provide I am inclined to avoid making any guess. Consider that we have no statement as to the intended output current, and the frequency is a guess based on the values of the timing components.