Hi, I looked for a birth/baptism for Wninfred and the only two I could find between 1810-1835 are the following;
Baptism:
Winifred Burke
baptised: 12 Feb 1818
place:Tuam,Galway
parents:Patrick Burke & Bridget Feely
* Bridget is not the mother your looking for
**also found a baptism for a Winifred born 1828 Galway to father Patrick but again mother is different-this time its Elizabeth Kiggins
Funny enough tho' I found 6 births/baptisms for children born to Patrick Burke & Mary Ward. They are;
Patrick
born 18 Feb 1814
bap 22 Feb 1814
Loughrea,Galway
parents:Patrick Burke/Mary Ward
sponsors:Stephan Madden/Clara Lynch
Maria Ann
born 7 Feb 1817
bap 12 Feb 1817
Loughrea,Galway
parents:Patrick Burke/Mary Ward
sponsors:Denis Lorcan/Prudentic Rigney
Catherine
born 14 Feb 1824
bap 15 Feb 1824
Loughrea,Galway
parents:Patrick Burke/Mary Ward
sponsors:Patrick Sweeney/Margaret King
Bridget
born 4 Dec 1826
bap 6 Dec 1826
Loughrea,Galway
parents:Patrick Burke/Mary Ward
Bridget
born 9 Oct 1828
bap 12 Oct 1828
Loughrea,Galway
parents:Patrick Burke/Mary Ward
Honora
bap 14 July 1831
Loughrea,Galway
parents:Patrick Burke/Mary Ward
Being as there is a gap between 1817-1824 either the records don't exist.are illegible, or missing. When the parish records come online this summer(fingers crossed) you could check Loughrea again to see what the story is between 1817-1824.
Also, I cold not locate a marriage for the parents in Galway or anywhere else during that timeframe. Hope this helps,Kim
** you may want to email the National Archives in Dublin as they hold the transportation records and also9 many of the court records and may be able to direct you as to where to get a trial copy.