I searched the Irish Family History Foundation's database. It contains 11,500,000 survivng rcds for abt 60% of Irish counties. Rcds B4 1840 are sparse.
Your Derrygolagh is likely the townland of Derrygowla. The two sound the same when pronounced. Derrygowla is in the RC parish of Islandeady.
For Thomas Higgins M 1871 +/- 4 yrs, I found 21 M rcds in Co. Mayo. For Ann Mulvanny (+variants) ditto, I found 0 M rcds. It may be that the search range isn't correct, the M event wasn't rcd'd, the rcd didn't survive, or rcd hasn't been digitized yet.
For Sarah Mullen B 1877 +/- 4 yrs in all Ireland, I found 47 B rcds. One of these MAY be for your ancestor.
For Luke Mullen M 1872 +/- 4 yrs ditto, I found 5 M rcds: 1869 Sligo, 1870 Galway, 1872 Galway, and 2-1876 Mayo. For Sarah Connelly ditto, I found 50 M rcds. Of these 50, only 2 M rcds correlate with those for Luke for both year and county. These are 1870 Galway and 1872 Galway. Will leave it to you to see if these rcds also match on the same parish.
B/M/D recording was civil after 1863, and therefore, fairly comprehensive. Check with the General Register Office at http://www.groireland.iehttp://www.groireland.ie for its rcds of these events.
Google the Foundation's name to lead you to this database to do more detailed research. Pls post your search results to my Genforum msg.