It was written by George Harrison with John Lennon as a parody of The Shadows style. (The Shadows, who backed Cliff Richard were the biggest British instrumental rock & roll group at the time of the recording.) It imitates the lead guitar with typical Hank Marvin licks, the melodic bass fills, and even has an imitation during the second middle 8 of the famous Jet Harris yell. It is the only Beatles track to be credited to Lennon and Harrison alone.
It was intended to be released as the b-side of "Why", another Sheridan song with The Beatles, but the record company chose to release another song instead. In early 1964, as The Beatles were gaining popularity, the record company Polydor decided to release it, with "Why" changed to the b-side. According to Bill Harry, editor of the Mersey Beat newspaper, Cry for a Shadow's original title was "Beatle Bop".