The song is “Haunted”.
It was written by Shane MacGowan, but the soundtrack didn’t have Sinead O’Conner on it, I think. Part of me thinks it was Kirsty MacColl, but another part of me thinks that is wrong.
My Google kung fu has been weak enough to stop me from answering this question for at least an hour now, and I can’t think how I would even find my cassette tapes from the era of the film, let alone motivate myself to dig them out.
Help me.