I did some research in Dublin a couple of years back about the name and the first time it surfaces was during the middle of the 16th century in County Donegal- I think it was south Donegal . At that time the Mullarkey family seemed to have the job ofsilversmiths to the O'Donnell family who were the rulers of County Donegal at that time. When I was doing my research I remember seeing a picture of a chalice which is in the care of a catholic religious order in Canada. As far as I remember the chalice was made for the the purpose of a wedding dowry involving a member of the O'Donnell family. What is more important though is that under the base of the chalice is inscribed in Latin the words "John O Mullarkey silversmith to O'Donnell made me" THe date was also given and as far as I remember it was 1560 or thereabouts. Around 1590 the Mullarkey name again crops up - chiefly in connection with a Fransciscan Friar whom believe was called Edward (or Eadhbhart in Irish ) Mullarkey. He was involved in various political intrigues which were rife at the time. He was I believe a close political advisor to O'Donnell. As far as I remember some trouble occurred and I think Edward was imprisoned for a short period - for what I dont know.
Around this time late 16th centurythe entire Mullarkey family seemed to leave their Donegal homeland en masse - whether they had to go ie were forced out by the O'Donnell family or were displaced by the English army of Elizabeth the First who were invading at the time we do not know. However some sources seemed to suggest that the O'Donnells wanted them out for some reason.
Out they went to the province of Connacht to the south. One group settled near the Tubbercurry area of County Sligo - others travelled to the the Dunmore area of County Galway. As far as I believe these areas still have the highest concentrations of Mullarkeys in Ireland.
In 1688-9 the Catholic King James came to Connacht looking for soldiers for his army to fight the Protestent King William. James recruited some Mullarkeys in Galway - I believe -and they marched off to the province of Ulster to fight King William's forces. Some of these Connacht Mullarkeys in James's army fought at the battle of Newtownbutler in county Fermanagh in 1689- however this battle went badly for the Mullarkeys and very many of them were killed. A small amount of these Connacht Mullarkeys managed to escape to the Roslea area of County Fermanagh which is about 10 miles from the battle - and began working as weavers and farmers. Their descendants ( of which I am one )are still to be found in this area.