You have two unusual names that may help your search, that being Barbara and Winifred. With a close birth year and first and last names you should be able to find parish baptism
records without a birth place/county. then you can search for further records based on the information. The surname for Michael will help as well as it is quite different from most.
You should be able to search Ireland with his birth year and his name only. You also have both parents name for Coleman and if he was born in Ireland it another easy name to search for. All the other children have very usual names.
There is no Conley's in the 1856 Galway Griffith Valuation but 68 Mulkerrin's, none in a place name Lettermullen but
Letterdeskert Townland is mentioned. Letterdeskert is in the Parish of Moyrus, Union of Clifden.
No Conley's in the phone book for Galway. Mulkerrin there are about 35 and 3 spelled Mulkern.
I would suggest starting with the Irish History Foundation mentioned above and then proceed from there.www.familysearch.org
Coleman Mulkerrin is not such an unusual name in Galway, some with parents names some without
those with a father Michael and wife Barbara did not have the surname Conley. Lee and Earles.