I can find no results matching Anna with a mother called Bridget born in Mayo around 1828.
Records for Ballycastle start very late, 1864, so there will be no hope of finding anything there.
Where did you come across the information that Anna was born in 1828 in Ballycastle?
Where did you find her mother was called Bridget?
How do you know her father died in 1868 without knowing his name?