Hello(I know that this is an old thread)
But I too agree with Horse.

I have taken a look at Ireland
Firstly I started with Mary Barrett born 1855 and worked back to Ireland
was able to find the following

JOHN WILLIAM BARRETT,born County Mayo 1832,he died Liverpool 1892,married a MARGARET MANGON born 1832 Ireland ,died Liverpool, April quarter 1893 and has far has I can tell they had Mary and 2 more children
MARY BARRET married WILLIAM DAVIES at St Peters Church,Liverpool in 1878
and I can only find a daughter Catherine Davies born Liverpool 1891 died the October quarter of 1918,there is a marriage for Catherine to Oscar Charleson at West Derby ,Lancs 0n the 25/7/1916 at St Peters Church Liverpool
***Also check out the following if you haven`t already this covers the Castlebar area of County Mayo which is where you need to be looking
castlebar District Births Register,County Mayo
www.from-Ireland.net/castlebar-birthrecords Mayo
Kushti Bok(good luck)
**Hope that is of help
take care
Romany Rosex