Hello Maine, I believe I can help you. All of the people you mentioned are in my family tree. My maiden name is Mulloy, and my ancestors were from Lavy More, Charlestown, Co. Mayo. Many of them emigrated to New York.
The link you provided doesn't lead to a specific document, so I can't view the one you're referring to, but I think you may have got mixed up. I will attempt to summarise below as succinctly as possible, but it's a bit convoluted so bear with me!
Dominic Mulloy (sr) was my 3rd great-uncle. He and his wife Bridget Brett had a son, Domnick (jr), born in 1872. I have no record of Domnick (jr) marrying or having children. It is possible that he had a daughter named Mary Agnes, but if she was born out of wedlock, there are unlikely to be any records for her.
I do have a Mary Agnes Mulloy in my tree, born 21st December 1905 (sometimes listed as 1906) to John Mulloy (Molloy) and Agnes (nee Fleming, widow Kilbride). John Mulloy was Dominic Mulloy (sr)'s nephew, via Dominic's brother, Denis Mulloy. So this Mary Agnes was Dominic (sr)'s great-niece, and Domnick (jr)'s 1st cousin once removed, rather than a daughter to either one of them. Could she be the one you're looking for?
Mary Agnes sailed to New York in 1923 on the Franconia, arriving on 3rd July. So the dates would appear to match up with your father's birth year, or close enough. I am assuming that your father took his mother's surname and you don't know who his father (your paternal grandfather) was? I don't have a record of this Mary Agnes's marriage or children, but if you can provide me with your paternal grandfather's surname, I may be able to find the records (all my info is from ancestry.co.uk).
I am also assuming that, since her surname was Brett, Katie Brett was from the Brett line and not the Mulloy one. So I don't have any information on her. I did a quick search on Bridget Brett to check for a possible link there, but there's almost no information about her. If Katie was the niece of Bridget Brett, that would make her something like Mary Agnes's second cousin once removed, by marriage. Eek, I'm getting confused! But it is conceivable that Mary Agnes referred to Katie as her cousin, even if she was a more complex relative.
It may be easier for you to just take a look at my Mulloy family tree on ancestry.co.uk. If you give me your email address I can send you an invitation to view it. I am attaching two photographs: one is of Mary Agnes and her siblings, the children of John Mulloy and Agnes (the 'Kate' in the caption refers to Catherine Mulloy, not Katie Brett). The other is of John and Agnes and three of their children (including Mary Agnes). There may be a family resemblance for you. I hope some of this helps!