Movie with a song

Hi. About 10 years ago I watched this american(?) movie about a single mother restlessly moving from town to town, which her daughter disapproves of. The mother makes her living from singing in small joints, at some point she sings a version of “Looking for the Heart of Saturday Night”. She falls in love, but she and the boyfriend are being chased, I don’t remember why. At the ending they are shipwrecked on a small ireland, and when a plane finally arrives we don’t get to know whether it’ll be their rescue or their doom. Can anyone help with a title or a tip? Regards from Copenhagen.

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