In 1841 Elizabeth AINSWORTH is living with Robert and Dorothy WHITTAKER in Darwen. There is also a boy - Henry AINSWORTH b abt 1839.
No mention can be found of this family in 1851.
In 1861, Betty HAWORTH daughter of Robert and Dorothy WHITTAKER is found in Oswaldtwistle with her sons Henry AINSWORTH b abt 1839 and Robert b abt 1859. On checking the LancsBMD site, Robert was born in Oswaldtwistle in 1858 and the mother's maiden name is AINSWORTH. No marriage can be found for Betty AINSWORTH to a HAWORTH so I suggest that HAWORTH may be a mistranscription on the ennumerator's part and that Betty HAWORTH is, in fact, Betty AINSWORTH.
I have been unable to find the family in the 1871 census, but in 1881 Betty AINSWORTH and her two sons HEnry and Robert are still living in Oswaldtwistle