I can locate some information on your family but will need some clarification from you. First, I was having difficulty finding any baptismal records for children of Daniel Murphy and Ellen Flemming on the subscription site Roots Ireland. I then went to the 1901 census and despite all the John Murphy records in Waterford, I was able to find the 1901 census record for the family which also included his brother Daniel. Both John and Daniel were born in Co. Cork which explained why I could not find any baptismal records since parishes in the Diocese of Cork and Ross are not on the subscription site Roots Ireland.
I then located the 1911 census record for John and Kathleen Murphy and then found a separate record for Daniel Murphy and his recent bride Nellie. I saw she was born in Co. Kilkenny and then located the 1910 civil marriage record for Daniel Murphy and Ellen Corcoran. See four links below.
There is a separate database on the free site www.irishgenealogy.ie which has the church records for parishes in the Diocese of Cork and Ross. There are baptismal records for children of Daniel Murphy and Hannah Fleming and other records for children of Daniel Murphy and Johanna Fleming. All of these records are in the Cork South parish also known as St. Finbar's. The earliest of these records is in 1864 so if the parents were married in 1844, there is a big gap from 1844-1864. I don't see a good baptismal record for John Murphy. He likely would have been born around 1855. I do see a January 29 1866 baptismal record for a Daniel Murphy with parents Daniel Murphy and Hannah Fleming.
So two questions. How many children are you aware of for Daniel Murphy and Ellen Flemming? Second, are your reasonably sure the 1844 marriage record you cite is for your ancestors.