When I do a search on "Pick Purnell" in the database of ASCAP (the American Society of Composers, Authors and Publishers, one of several performing rights organizations in the USA), up come listings for 29 songs, including "Sit This One Out", all credited to one Andrew J. Kaulkin.
Andy Kaulkin is the producer for Solomon Burke's album Don't Give Up On Me. So it appears that Kaulkin wrote that song for Solomon Burke but used the pseudonym "Pick Purnell" in the album credits.
Andy Kaulkin is also the president of a large and busy independent record label called ANTI-, a division of Epitaph Records, which released Burke's album.
ASCAP lists an address for the publishing business Andy Kaulkin Music, in Calabasas, California, and gives a telephone number and email address.
Andy Kaulkin Music is administered by Wixen Music Publishing. There is no information about Andy Kaulkin or Pick Purnell on that website, but perhaps you could write to them and ask.
Allmusic.com lists one album for Andy Kaulkin, Six Foot Seven and Rising, released in 1996, and described as a blues record.
The Andy Kaulkin album is for sale as MP3 downloads or as a used CD at Amazon.com.
One review on Amazon says that "Andy Kaulkin has been a seasoned L.A. musician for decades... Andy is an incredible keyboard player, but I especially like his fast and hot boogie-woogie style."
I found a photo of him here, dated 2005.
Bottom line: If you want to find out about a songwriter in the USA, there are two places to look first: search the databases of ASCAP and BMI.