I have twins, Ann and Thomas born in that same time frame to Richard and Mary Farnworth. Richard Farnworth appears in the 1841 census living at "the Barracks", Clifton, Lancashire, England.
I also have in my notes a reference to a Lawrence Farnworth from 1722 being married to Alice Baxter, 1724. I believe the years referenced are birth years. These references are from in and around the Eccles, Lancashire parish church, St Pauls.
This may not be your Ann Farnworth as the census shows Ann's mother as Mary born around 1799-1800.