Hi there,
Do you have any idea where in Cork they might be from? Cork is a very big county with many, many parishes. Without a parish, I cannot direct you to church records. If you knew, you could then contact the parish priest to assist you in your research.
Civil registration records start from 1864 and are available from the General Register Office (GRO) here: http://www.groireland.ie/research.htm
You could try checking the land records called the Tithe Applotment Books (1823-38) or the later Griffth's Valuation (1848-64). Griffith's is freely available here: www.askaboutireland.com or here: www.failteromhat.com Failte Romhat has lots of other useful links you could try looking at.
If you are carrying out research, you should also remember to check the variant spellings of Hagarty, such as Hegarty, etc. You can see the frequency of the name Hagarty here: http://www.irishtimes.com/ancestor/surname/index.cfm?fuseaction=Go.&UserID=
You should check any immigration records in NZ, as generally most information was given at the port of arrival rather than the port of departure.
Please make sure you link anyone else in your family who is interested in their Irish heritage to our site - and indeed anyone else you know of Irish heritage.
Kind regards,
Sinead Cooney
Genealogist (Ireland XO)