Obvious characteristics of a damaged capacitor are leakage of a brownish fluid, corrosion, or severed leads. Non-obvious signs could be measured with a capacity gauge (for the professional handyman).

Search for the pegs of the capacitor on the PCB and warm up the soldering iron. Remember orientation (left-right)!

The capacitor specification should be: 470nF, min. 275 VAC, 22.5 mm pitch. Only use X2-Class qualification which can be distinguished on the part. Capacitors come in blue and yellow colors.

Caution: the soldering iron will be hot. Perform this action in a well-ventilated area.