Finding a drama thriller B&W movie from the 40's or 50's

A movie I saw on TV as a child–a young girl is the protagonist. An young handsome Irishman shows up as handyman of some sort and befriends her. He is nice to her but could be a bad guy. In one scene he is singing and explaining the song Molly Malone, that you have to sing “Streets broad and narrah” for it to rhyme with “wheeled her wheelbarrah.”

a tree grows in brooklyn?