Hi Sue,
Thanks for your message.
There are 3 Francis McDonnell?s recorded in CO. Mayo in the Tithe Applotment Books:
http://titheapplotmentbooks.nationalarchives.ie/search/tab/results.jsp?surname=McDONNELL&firstname=francis&county=Mayo&parish=&townland=&search=Search
The TIthes are land records and also though they do not give any genealogical info they can sometimes be used to tie a person to a specific place. These parish names may be a starting point for you if all else fails.
You could also try searching online for records. There are a number of places to do this including:
www.familysearch.org has a huge database of genealogical records including some church records for Ireland.
www.rootsireland.ie have a large collection of Irish Church records, however you have to pay to use this site.
http://www.mayoancestors.com/default.aspx have records from Mayo, however you have to register to see the full record.
Some other websites that may be helpful are:
The National Archives of Ireland http://www.nationalarchives.ie/genealogy1/genealogy-records/introduction/
The National Library of Ireland http://www.nli.ie/en/family-history-introduction.aspx
The National Archives UK ? genealogy search: http://www.nationalarchives.gov.uk/records/looking-for-person/
The Public Records Office of Northern Ireland http://www.proni.gov.uk/index/family_history.htm
Irish Times: http://www.irishtimes.com/ancestor/index.htm
Irish Genealogy Tool kit: http://www.irish-genealogy-toolkit.com/
Genealogy Links: http://www.genealogylinks.net/uk/ireland/mayo/index.html
Clare Library: www.clarelibrary.ie/eolas/coclare/genealogy/genealog.htm
Mayo: www.mayo.ie
Remember to post any new information that you find here. 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.
Kind regards,
Genealogy Support