The scheduled one-off gig was scrapped last week after Morrissey collapsed after a show in Boston which he blamed on "a horrendous cold" he had caught from support act Kristeen Young.
Sir Cliff said the show next Saturday is to thank his "loyal fans" in America and will include a question-and-answer session as well as performances of hits such as Devil Woman and We Don't Talk Anymore.
He said: "I know that many fans were disappointed about the Morrissey concert being cancelled and I was disappointed as well.
"I've decided I'm going to go anyway to New York and I'm going to put on an evening. While it won't be the show I had planned to do with Morrissey, it will be a really nice, intimate evening with my fans."