You mention two children. I found records for some more on Roots Ireland.
James 6 Dec 1818. Mother is listed as Helen Connors.
Catherine 28 April 1821(Helen Connors)
John 14 Nov 1825 (Eleanor Connors) This one is at odds with your record of Michael born 3 Nov 1825 but I have had a look at the register and to my mind it is Joanem or similar and the figure that looks like a 3 is actually the B for Baptised.
Regarding Andrew Wyse, is he your direct ancestor? Do you know for sure that he is the child of Michael and Ellen?
On Find my Past there is a valuation office record of Andrew in Ballycourdra in 1850 so, putting in doubt that it is the same Andrew as appears in the census in New York the same year.
If Andrew is the son of Michael and Ellen it is likely he was born between Mar 1816 and June 1818 where there is a gap in the parish registers. But the trouble is, without this record it is impossible to prove that he is their son unless you have other information such as a death record that shows his parents names.
I am curious to know what record you have that shows Ellen in 1863. Why do you estimate her death as after 1893? If this is the case there should be a record of it in the Civil Registrations and there does not appear to be any such record.
I did however find another record on Find my Past with reference to Ellen Wyse. This is from the 'Waterford Poor Law Union Board of Guardians Minute Books' It is dated 14 Feb 1856 and is a request by a Mr Laurence Power of Ballytruckle requesting his name be placed on the Rate Book for the house office and land formerly held by Ellen Wyse.
I don't know if this means that Ellen had died or had simply moved off the land.
The alternative Parish name is simply that, another name for the parish. In this case St John's was/is also called Ballygunner. There are no separate records, they are the same thing.