CapeCollen, there was another Bernard Heavey died 1892, maybe you can do this look-up also. Bernard b. 1896 this must be Michael & Anna's son and Bernard 1909 John & Ellen's son.
I found some information on Michael & Anna's children born in England. Bernie, you can send for these birth certificates and they might give you some more information. I haven't been able to locate them in the 1881 census yet and they are gone before 1891 according to their US census information. The children were registered in Nottingham, Nottinghamshire, England as follows:
John William HAVY 1881; Patrick Bernard HEAVEY b. 1883 d. 1883; Catherine Lizzie b. 1884 d. 1887; Bernard 1885; James Lawrence 1887, d. 1888; Bridget 1888.
1900 census for Michael & Anna Heavey is on ancestry.com as HASREY. They state 18 children-6 living. Maybe CapeColleen can look-up their oldest child born in the US, Mary September 1889 to see where they were living, next child Amelia was b. CT 1895, the Bernard 1896 MA and Sarah April 1898.
I also found Nellie Cronin's family in Ontario, Wayne County, New York inthe 1880 census. The records for Ontario, NY at their town hall begin with 1882 as stated on their website. Patrick & Kate Cronin state two daughters b. in Michigan; Kate 12 & Mary A 10. When I look at Michigan for 1870, the best family I find is in Negaunee, Marquette, Michigan but it looks like Kate Cronin might be married to Patrick's brother and Patrick is living with them as a boarder.
I'll keep looking at the English census records to see if they can't be found since John was born supposedly Jan 1881. These census records sometimes give the county in Ireland as a place of birth. Also since Anna was traveling with Bridget & John as children, it might be easier to find them on the immigration records.
Happy New Year to All,
Cheryl