Unfortunately you need to know where to look i.e. townland or parish, so you can check what records are available for the area that your Hannah was baptised.
Cork is the largest county in Ireland with over 100 RC parishes (assuming she was Catholic), and not all of these have records going back to 1850.
The parents names could be a very useful clue in confirming, or ruling out, any records you find - but you have to find possible matches first..
Parish records for part of Co. Cork, mostly south west, are available free on
http://www.irishgenealogy.ie/ Transcripts of other Co. Cork records, for the north east of the county are available on the pay website of the Irish Family History Foundation ad :
www.rootsireland.ieDo you have her father's name - you didnt mention it ?
Shane