That is a double socket. A plug is what fits into it. If the socket has no power then the circuit breaker at your consumer unit may have tripped (or fuse blown, if it's an ancient consumer unit).
I'd leave it as is if the runners can easily be removed when necessary. Any chance you could modify the CU lid or its fixings slightly to make it removable (but obviously securely closable too)?
The black cylinder is an electrolytic capacitor, which looks to have overheated (due to a high ESR?), judging by the underside of the board. What are its markings? It's certainly worth replacing; but the fault may lie elsewhere.