might be an idea to try following up one of the children births to confirm the mother's name.
All the locations mentioned for places of birth for Michael, Elizabeth and the children i.e. Carrigaline, Ballincollig, Ballygarvan & Passage west are in the Cork registration district. If they married in this area then there only appears to be one likely marriage in this district between 1888 and 1892. A Michael Foote, also listed as Michael Footte.
these are the references :
Name: Michael Foote
Registration district: Cork
Event type: Marriage
Quarter and year: Jul-Sep 1890
Volume: 5 / Page: 115
see :
Ordering Certs from GRO Roscommon Details included on a Marriage CertShane