William Corcoran c1843 - 1908

I am researching a William Corcoran (c1843 - 1908).  I have found him marrying an Ellen McHale in Castlebar Co Mayo on 28 June 1864.  I have William & Ellen in various census returns in Liverpool (1871, 1891 & 1901) and each time it just says 'Ireland' as his/their place of birth.  I have found a possible baptism for Ellen McHale on 24 September 1843, parents Thomas McHale and Bidy McLyons. 

I am trying to locate the birth/baptism of William Corcoran so I can identify his parents

I would appriciate any help the community could give.

Thank you.  Graham