Do I need to connect capacitor across MOV?
MOV is a voltage controlled R with some small C (pf).  For best transient immunity both clamping and C voltage divider methods are needed for ESD with sub nanosecond rise times are faster than the diode clamps can respond or if lightning ionization via line for same reasons, so additional bandwidth reduction methods are used such as C divider, LC LPF and clamps.

ESD is more likely if you have a floating plug and lightning when connected.

If you do not, and the ESD ionization arc time is faster than the MOV can respond and then it arcs over each subsequent parts rated for only xxx V then it will break your device.