There are a number of Anthony Staunton/Stantons in Belmullet in civil records on irishgenealogy.ie. Here is one of them:
Anthony Stanton, Caramore, Belmulllet, b 1883
Parents Patt, Catherine Heraghty
Grandfather James Heraghty
Your Anthony and Bridget had a child as early as 1901, so this Anthony would have been 18 then. Have a look at rootsireland; you should be able to find some records for your family.