A Little Thing Called Murder (2006 TV movie) starring Judy Davis, Jonathan Jackson, Chelcie Ross
IMDb synopsis: Based on the true story of mother/son tag-team Sante Kimes and her offspring, Kenny, who crisscrossed the country and committed a string of crimes, among them robbery, fraud, arson, slavery, and murders that shocked the world. Sante Kimes found a rich man, Ken, but he wasn't dumb enough to marry her. So her son Kenny Jr. isn't co-heir with his stepsiblings. However, the boy is made her accomplice in an endless series of shameless crimes, from shoplifting to elaborate scams. After her four year jail term, Kenny is unable to escape her clasp again. When his father dies, he's dragged along throughout the US as an 'enforcer' for even more daring crimes, as she's determined to avoid jail by murdering dangerous witnesses.
This is a bit of a guess, as I have only a fragment of memory. But, I was thinking that Sante sings that song. There is a second movie about them: Like Mother Like Son: The Strange Story of Sante and Kenny Kimes (2001 TV movie) starring Mary Tyler Moore, Gabriel Olds and Robert Forster.