Tom, It is a rarity to see the County of Cavan along with the Mulgrew name. Very definitely an Irish name and definitely a Tyrone name. If you look at the name over the past two hundred years you will find that it mostly emanated from the Stewart estate around Ballygawley - Errigal Keerogue parish - Ballymacelroy townland. Over the years they spread in a very narrow band Eastward through Dungannon, Cookstown, Coalisland, Stewartstown. From this area of Mid Ulster there are very few incidences of the family name until you get to Belfast to where they followed the Linnen trade (jobs). You will also come across small pockets in Mayo, Banbridge, Limavaddy and Lissan. Nearly all of these pockets can be traced back to Ballygawley District.
The only incidence of Co. Cavan that I have come across is when my fathers uncle married a lady from Cotehill, Co Cavan and then went to the USA to live.