Hello:
I see that there is a church baptism record for Henry Barry from Bunclody parish in Wexford baptised on 27 October 1811 to John Barry and Esther Barry by Rev. Alex McIntosh. This is a Church of Ireand record. The transcription is available to view on Roots Ireland which is a subscription site. I did not see any other children born to this couple from Bunclody parish registers.
I checked for a William Barry/Berry and a Margaret Byrne family but was not able to find any transcriptions from the registers available on the above site.
You might consider creating an Ancestor profile for this family through the XOChronicles feature which is located at this link:
https://irelandxo.com/ireland-xo/history-and-genealogy/ancestor-database
If you have any further questions, please let us know.
All the best,
Jane