J.K. Rowling named Argus Filch's cat Mrs. Norris after the character in Mansfield Park as a nod to Jane Austen, who is one of Rowling's favorite authors. (Source)
Jane Austen actually knew a lot about the professions featured in Mansfield Park. She was the daughter of a clergyman, she had a brother in the clergy, and had two more brothers who were in the navy. (Source: Introduction. Mansfield Park. By Jane Austen. Bantam Classic Edition. 1983.)
William Cowper, the poet that Fanny references, was actually one of Austen's favorites. Cowper was a forerunner to romantic poets like William Wordsworth and Samuel Coleridge, also read by Austen. (Source: Michael Karounos. "Ordination and Revolution in Mansfield Park." Studies in English Literature, 1500-1900. September 2004.)
Conservative philosopher Edmund Burke is noted by many critics as a major influence in Mansfield Park. Burke used a metaphor of a country house to describe his ideal, conservative system of government. Austen, of course, uses lots of country houses in her book, and she was familiar with Burke's political ideas. Perhaps the conservative Edmund Bertram is a shout-out to Edmund Burke? (Source: Michael Karounos. "Ordination and Revolution in Mansfield Park." Studies in English Literature, 1500-1900. September 2004.)
Lovers' Vows, the play performed in the book, was a controversial political play at the time Austen wrote Mansfield Park. It was translated from German to English by Elizabeth Inchbald, who was a prominent sympathizer with the French Revolution. England as a whole was sharply divided politically over the French Revolution: liberals liked it and conservatives hated it, for the most part. (Source: Claybaugh, Amanda. Introduction. Mansfield Park. By Jane Austen. Barnes and Noble Classics, 2004.)
Interestingly, two of the most recent theatrical adaptations of Mansfield Park (the 1999 movie and the 2007 Masterpiece Classic production) change the character of Fanny around a lot. In both movies, Fanny is much more outgoing and spunky than she is in the book.