What I'm talking about is what Bruce Springsteen does to the classic song "Santa Claus is Coming to Town". His version of the song doesn't have the same melody as the original. Instead it has a flattened melody. I've heard pop and rock singers do it to other songs, so there must be a term for it. Does anybody know?
They are "interpreting" the melody.
It's like in jazz and lounge singing when the singer doesn't sing the melody as published.
To me it seems the thing most often/noticeably change is the melodic contour, but the the rhythm is more or less kept the same. "Flattening" out is changing the melodic contour. Of course it isn't flattened in the sense of accidentals (flats and naturals) it's a change of the direction of the line.
It exploits the fact that many melodies are composed of chord tones and you can substitute one chord tone for another without changing the harmony of the song. If you had a melody ascend a chord like
G B D G you could just sing
G G G G and it will still match the harmony. If you sing the line with the original rhythm, it can still be recognized as an "interpretation" of the original.
You can judge whether Springsteen's interpreting is as good as Sinatra or Billie Holiday, but the essence of what is happening is the same.
Yes Springsteen is covering the Crystals version (see @Tetsujin's comment). But most of the original melody (hardly a folk song BTW, composed in 1934 by the wonderfully-named J. Fred Coots) survived into their recording, they just re-thought the 'Santa Claus is coming to town' refrain. Springsteen only vaguely follows the shape of the melody throughout the song.
And yes, 'Flattening out' is a good description of what Springsteen does to a melody. I don't think there's any other generally-accepted term for it. His style (and that of many rock singers) is to shout everything out at the top of his range. Here's an interesting analysis by a vocal coach: