I assume that Errew where Michael lived and was possibly born was the one in Crossmolina civil parish, Tirawley barony, Baiina Poor Law Union in County Mayo. According to
https://www.swilson.info the linked R.C. parish is Crossmolina in Killala diocese, with church/chapels at Kilmurra (1785) and Crossmolina (1806). National Library of Ireland website has microfilms Crossmolina baptism registers online for August 1831-August 1841 and April 1845-December 1880.
Another Errew in Mayo is in Balllyhean civil parish, Carra barony, Castlebar PLU. Linked R.C. parish Eglish (or Aglish), Ballyheane & Breaghfy in Tuam Archdiocese. Churches/chapels at Castlebar and Ballyhane. (Btw those are all variant spellings of Ballyhean.)
There are also 2 places called Errew in Co. Leitrim.
A few more things to consider.
Michael may have lived in Errew as a child but may not have been born there. There was a custom for a woman to have her 1st baby at her mother's home. Maternal grandparents would arrange for priest at their church to baptise baby. Michael's mother may have been staying with relatives for some other reason when he was born.
Michael's father was a carpenter. He might have gone where he could find work. Wife & children may have gone with him, remained home or stayed with relatives. People went back & forth across Irish sea. Check birth registrations in Britain, just in case. They began 1837 in England & Wales, 1855(?) in Scotland. GRO index (England & Wales) is free to search. Obvious place to go from Mayo was Lancashire. Lancashire BMD is a volunteer site and is not complete but you can search as many years as you want at one go. Mother's maiden names are being added to GRO and Lancashire BMD. Lancashire Parish Clerks is another volunteer site which has added some R.C. parish registers recently. Manchester & Lancashire Family History Society site has an index to some Manchester Catholic registers. Other free sites are FreeReg, FreeBMD and of course Family Search. Merseyside and Midlands are 2 other regions with high Irish populations. Look for Michael and/or his dad on English census 1851, 1861 & 1871. Members of my Irish families went to & fro, so could turn up on an English or Irish census or be missing. One woman had alternate children in both countries until finally settling in England. There's even 1 Mayo family who went to U.S.A. for a while, had 2 sons there, then returned across the ocean and remained in England.