Wikipedia defines an answer song
as "a song (usually a recorded track) made in answer to a previous song, normally by another artist." For example, "Yes, I'm Lonesome Tonight" by Dodie Stevens was the answer to Elvis Presley's "Are You Lonesome Tonight?"
Frank Hoffmann and William Bailey, in their book Arts and Entertainment Fads
Answer songs were released in particularly large numbers during 1960-1962. The impetus for this deluge of recordings appears to have been the notable success of Jeanne Black's "He'll Have to Stay" (Capitol). Based upon Jim Reeves' "He'll Have to Go," Black's record entered Billboard's "Hot 100" on May 2, 1960, peaking at number four.
In this case, there was even an answer to Jeanne Black's answer: "I'm Gonna Stay" by Johnny Scoggins.
The Original: "He'll Have To Go" (by Jim Reeves, 1960)
The Answer: "He'll Have To Stay" (by Jeanne Black, 1960)
The answer to the answer: "I'm Gonna Stay" (by Johnny Scoggins, 1960)