Hello Rita,
The place in County Mayo that you call "CONNICK" sounds a lot like the phonetic spelling or pronunciation form for "Connacht" (or "Connaught"), the western province of Ireland that is comprised of the 5 counties of Mayo, Galway, Leitrim, Roscommon and Sligo.
There was a good deal of movement in the hard times of the 17th and 18th century---as in "To Hell or Connaught"---so maybe a person just gave "Connaught" as their place of origin to give people a rough idea of where they came from, without getting into a detailed listing (or without giving out too much information if they were worried about the possibility that the British were looking for them).
Many, many O'Malley/Malley names in County Mayo, as well as elsewhere. Don't forget that you can sign up for the "ancestry.com" website for free for the first two weeks and then cancel out your account just before the two weeks are up. And The Latter Day Saints (Mormon) library's website at "familysearch.org" is free and loaded with information after you join (...joining just the website, not the church itself).
Good luck.
----Robert O'Keane