“Leave My Kitten Alone” is indeed a cover song from The Beatles, but it is available on their 1995 album, “Anthology 1”.
In Detail
Album
Recorded: 14th August, 1964
Studio: EMI Studios, London
Genre: Rock and roll
Track Duration: 2:57
Record Label: Apple Records
Songwriters: Little Willie John, James ‘Jay’ McDougal, Titus Turner
Producer: George Martin
Engineer: Norman Smith
Performers

Anthology 1 Album (1995)
Paul McCartney: bass, piano
George Harrison: lead guitar
Ringo Starr: drums, tambourine
Track Source
Leave My Kitten Alone
In 1959, Little Willie John managed to get to number 13 on the American R&B charts with this song, but in the pop charts, it only got to number 60. Johnny Preston then covered the song in 1961 which reached number 73. None of these singles got to the UK charts.
Like most budding bands, The Beatles would sing many cover songs in order to gain popularity. On many occasions, they would sing some obscure songs but they had the knack of sprucing them up. In this case, while the original version by Little Willie John was a soul recording, The Beatles sang it in their own rock and roll style from time to time:
“That was a Johnny Preston song that we’d rehearsed in Liverpool along with all our Cavern stuff and it was just in our repertoire. It wasn’t a big one that we used to do, we’d pull it out of the hat occasionally, and we also recorded it.”
Paul McCartney
The Complete Beatles Recording Sessions, Mark Lewisohn
Although earmarked for inclusion on the “Beatles For Sale” album, for some reason they dropped the idea.
Recording Studio
The Beatles began recording “Leave My Kitten Alone” on the 14th August, 1964, in Studio Two at the EMI Studios, London during the 10.00pm-11.15pm session. They recorded five takes with the last of those being suitable as the rhythm track for overdubbing.
After overdubbing extra vocals and a guitar piece, they more or less forgot about the track. In fact, they didn’t even get around to mix it properly. However, we finally saw it surface on their 1995 album, “Anthology 1”. There is also a bootleg version from a 1985 stereo mix which didn’t become commercially available.