Karen:
First of all, I am very excited to hear from you. What is your interest in my great-great grandfather, James P. Tharp? Are you kin (hopefully)?
I am not unaware of his census claim to have been born in Kentucky and I cannot give a definitive answer concerning his place of birth. I have searched unsuccessfully to locate a place of birth in Kentucky or even any mention of his parents' names.
My source is our family bible which belonged to my great-grandmother (his daughter) Emma Jane Tharp Bodkin. In it was recorded his date of birth (March 3, 1844) and date of death (December 14, 1895). Grandma Bodkin specifically mentions "Salt River Township in Audrain County, Missouri" as his place of birth, but I have no way of knowing where she got her information.
To be honest, I myself, tend to believe that he might very well have been born in Kentucky since there are few if any Tharps that I've found living in Audrain or Callaway counties in the early 1840s.
To date I have had to content myself with his place of birth being "either Salt River Twp, Audrain Co., Missouri" or the "state of Kentucky".
Please, let me know how (or if) you connect to this family. I so want to locate other surviving relatives.

Rod Green