There is one Dolphin in the Sligo telephone book, it might be
worth writing to this person.
1901/1911 Census: www.census.nationalarchives.ie
17 Dunneil, Dromore, Sligo
Thomas Dolphin age 50 has a son Michael S. age 2
Mary Ann 30 married 2 years, 2 born, 2 living
Mary Ellen 1
Most county genealogy offices do not go up to 1935 so you can try looking for the Marriage of a Michael in this records: 1864-1950'swww.familysearch.org
(usually more information)
Francis Dolphin born Jan-March 1938 no parents names given
You could try Naturalization record for Michael Dolphin father as it might give his parents names and a place.