I looked previously for Ann Kavanagh in the Liverpool area and the only one I have found so far who is a possible match is :
1881 3689/5/3
Kirkby
Ann Cavannah, 28 yrs, Domestic Servant, b Wexford
Added
There is a death registered in West Derby district in 1888
Ann Kavanagh 33 yrs
Burial at Ford Cemetery with an address of 18 Carnarvon Street but difficult to say if these are the same people and also your family.
I know this tricky I know Kirkby its outskirts of the city. Ford is a Catholic burial ground. I may look for her on 1881 census electrical rolls too