Hi Wendy
My last post has disappeared.

The baptism that Judith found on IGI for Arthur b. 1838 is more than likely the same person as the birth found in 1838 on free bdm.
After searching both the 1841 and 1851 Census I feel this is Arthur's baptism.
MARTIN Arthur
Baptism 26 Jul 1836 Suffolk England
Father Thomas MARTIN Mother Lydia
1841 Census HO107/1041/16;
Living at Market Wickham, Suffolk
MARTIN Lydia 35 Independent Not born county
MARTIN Lydia 11 Not born county
MARTIN Arthur 5 Born countyMARTIN Henry 3 Born county
MARTIN Horace 2 Born county
MARTIN Fanny 3 months Born county
MARTIN Fanny
Baptism 13 Aug 1841 Wickham Market, Suffolk Eng
Father Thomas MARTIN Mother Lydia
1851 Census HO107/1827/582
Living at West Rudham, Norfolk
MARTIN Thomas Head Married 70 years
Curate of West Rudham b. Edkirk, Norfolk
MARTIN Lydia Wife Married 48 b. Fakenham, Norfolk
MARTIN Lydia Dau Unmarried 21 b. Fakenham, Norfolk
MARTIN Arthur Son 15 b. Saxmundham, SuffolkMARTIN Henry Son 13 b. Saxmundham, Suffolk
MARTIN Horace Son 12 b.Saxmundham,Suffolk
I think the marriage in 1835 was possibly a second marriage for Lydia and Arthur their first born in 1836. We have parents with the correct names and the father's occupation is also correct.
I can't help you with searching for Catherine BARRY's birth or family in Ireland so am wondering what you would like to do now. Are you using a family history program to record the information?
Cheers
Cando