To begin with, “I Got A Woman” is a Ray Charles’ composition but The Beatles recorded their own version of it. In fact, they done two versions of the song for their shows on BBC Radio.
In Detail
Albums
Recorded: 13th August, 1963
Genre: Blues, rock
BBC Radio Show: Pop Go the Beatles
Songwriters: Ray Charles, Renald Richard
Producer: Terry Henebery
Performers
John Lennon: vocals, rhythm guitar
Paul McCartney: bass
George Harrison: lead guitar
Ringo Starr: drums
Track Sources
Live At The BBC
On Air – Live At The BBC Volume 2
I Got A Woman
Ray Charles released this song as a single in December, 1954 together with the B-side, “Come Back Baby”. Elvis Presley also recorded the song in 1956.

Live At The BBC Album (1994)

On Air – Live At The BBC Volume 2 Album (2013)
The Beatles’ Version
The Beatles often covered songs that came over to the UK from across the pond. The Fab Four perfected a way of breathing new life into some of those songs, in particular, the more obscure ones. In this case, they sang “I Got A Woman” many times in the early 1960s.
During their formative years, The Beatles were regulars on BBC Radio programmes and they recorded the song twice for the BBC. Since the Fab Four’s versions are similar to the Elvis Presley style, it is possible that they heard his copy first. In short, this was a track on the debut album “Elvis Presley” (the UK release was “Elvis Presley Rock n’ Roll”).
They recorded this song on the 16 July, 1963, at the BBC Paris Theatre in London for airing on the “Pop Go The Beatles” radio show on the 13th August. This version is now available on the 1994 album, “Live At The BBC”.
On the 31 March, 1964, the band recorded another version at the Playhouse Theatre in London. This version is slightly shorter than the previous recording and this was for airing on the BBC’s “Saturday Club” programme. This recording is now available on the 2013 album, “On Air – Live at the BBC Volume 2”.