I got the publication date wrong - this was one of Greene's later works, published in the early 80's. I remember reading (and loving) The Heart of the Matter, The End of the Affair...
Here's the premise of Monsignor Quixote...I think I'll read it this evening...it isn't very long.
"The title character of Monsignor Quixote is a village priest, elevated to the rank of monsignor through a clerical error, who travels to Madrid accompanied by his best friend, Sancho, the Communist ex-mayor of the village, in Greene's lighthearted variation on Cervantes."