Lady Catherine says that she recommended Miss Pope to Lady Metcalf as a governess after she was “was merely accidentally mentioned” to her. It doesn’t sound like Lady Catherine knew her at all, but Lady Metcalf reported she was “a treasure.”
From this information as well as historical trends, we can guess that Miss Pope is a gently-raised young woman who is descended from an impoverished family and/or nearing spinsterhood. She likely dwelt either in Kent or London prior to her employment with Lady Metcalf.