Anne is one of five children of Samuel (& Sarah) Pearce, a minister in Birmingham, who was friends with William Carey. See "The Seraphic Samuel PEARCE" https://banneroftruth.org/us/resources/articles/2001/the-seraphic-s... and https://www.wmcarey.edu/carey/gsmith/viii.htm. Samuel died of TB at age 33 in 1799, but two of his children (William and Anne) later followed his dream and became missionaries in India. The Carey-Pearce grandson, Rev. Samuel Pearce CAREY, wrote a biography on his great grandfather, William Carey. I'm not sure Anne's middle name is Hannah. I think it is Hopkins.
(More research is indicated for the exact location of the Carey-Pearce marriage because the mission was in Berhampore which was a Danish settlement out of the control of the East India Company who controlled India until 1858.)