Although the fugitive and the sergeant are on opposite sides of the law, they're both Irishmen. They come from similar backgrounds and share a number of cultural references that, under normal circumstances, would probably make them the best of pals. But this isn't a normal situation. In colonial Ireland, the police force consists largely of native Irishmen, who enforce the rigors of English law against their own people.
But underneath the officer's uniform there still beats an Irish heart, and the fugitive uses this to his advantage by playing on the police officer's fondness for an old sentimental Irish ballad. The ballad reminds the officer of his distant youth, long before he agreed to sign up for the police force and do the colonial authorities' bidding. This makes him question his life choices, possibly for the first time since he joined the police force.
By reminding the officer of his indomitable Irishry, the balladeer fugitive is gradually reestablishing the connection between the officer and his people which was broken on that fateful day when he decided to join the police force. Though he's angry when he finds out he's been duped, the officer nonetheless lets the fugitive go, a clear indication that he's not quite the same person as he was before. He may be much less of a police officer than he once was, but he's also a good deal more Irish.