Cartwright was taught at Queen Elizabeth Grammar School, Wakefield, University College, Oxford, and for an MA degree at Magdalen College, Oxford, (awarded 1766) where he was received a demyship and was elected a Fellow of the College. He became a clergyman of the Church of England. Cartwright began his career as a clergyman, becoming, in 1779, rector of Goadby Marwood, Leicestershire. In 1783, he was elected a prebendary at Lincoln Cathedral.