Hae you tried other versions of the name?
I don't have anything much as early as this, but the most popular spelling seems to be Mullarky.
I have found
Mullarkey, Mularkey, Mularky, Milarky, Milarkey, Molarkey, and Malarky, Malarkey.
You can see the different spelling on the Mormon Site.www.familysearch.org