On the 1871 census Elizabeth gives her place of birth as "Iregaen, Wales". There is no such place.
The first letter has to be a "T", so I think it could be the parish of Tregaean/Tregaian, near Llangefni in Anglesey?
Note that they have a lodger, Hugh Hughes, of the parish of Llanbadrig in Anglesey living with them.
The family lived in Everton and had most of their children christened at St. Peter's Parish Church. These records suggest that they had two sons named Owen Rowland Davies. The first was born on 13 Feb 1860 (but had probably died before the 1861 census?), the second in 1869.
Unfortunately the christening records do not give the mother's maiden name, but you should be able to get this from any of the birth certificates: John Owen Davies (born 4 Aug 1861); Hugh Rowland Davies (born March 1863) or Hannah Jane Davies (born 26 May 1867).