Hi Kate
Thank you for your query. Do have any other information about John Joseph? What was his wife's name? children? etc. Remember to post as much information as you can with regard to the people you are researching. The more information you post, the more likely it is that one of our volunteers will be able to advise or assist you. Also include information concerning which sources you may have already used so others may further your search.
You have good information to start with but finding a place of origin in County Cork ( Parish or Townland) will allow you to search what Roman Catholic records still exist for that area. It may lead you to finding other ancestors or even making a connection with living relatives.
Generally, more information was given at the port of arrival rather than the port of departure. If you knew which city they/he arrived at, this could be a good place to find more information -and perhaps even find out an exact place of origin. There are lots of books and documents available about the Irish emigration to NZ. There is a website you could look at: http://www.teara.govt.nz/en/irish/2 for information about the Irish in New Zealand.
You could search for a record of John Joseph/ Thomas in the Tithe Applotment Books (1823-38) http://titheapplotmentbooks.nationalarchives.ie/search/tab/home.jsp or the later Griffith's Valuation (1848-64) http://www.askaboutireland.ie/griffith-valuation/ or the Church of Latter Day Saints (LDS) https://familysearch.org/
The Tithe Applotment List might be of use to you, or at least interesting for you. These lists constitute the only nationwide survey for the period, and are valuable because the heaviest burden of the tithes to the Established Church, the Church of Ireland, fell on the poorest, for whom few other records survive.
Civil registration records are available from the General Register Office (GRO). These start from 1864. You can access the website here: http://www.groireland.ie/research.htm
Here are some other websites that may be of help to you:
http://www.irishtimes.com/ancestor/browse/
http://www.corkpastandpresent.ie/
http://www.goireland.com/genealogy/county-centres/locations/cork-county…
http://www.skibbheritage.com/
http://www.corkarchives.ie/genealogy/
http://mallowheritagecentre.com/
All the best in your research Kate
Kind Regards
Genealogy Support