You should post on the England board rather the Wales board.
However, there is no birth record in England in the period 1870-1872 for a William Henry Harrison Kirkman.
There is a record for a William Henry Kirkman in Bury, Lancashire in the Oct quarter of 1871.
There is also a record for a William Henry Kirkman in West Derby, Lancashire in the Apr quarter of 1872.
In the 1881 Census there is a William H Kirkman aged 9 living with his parents and siblings in Manchester. His father is William and his mother Catherine. His siblings are Albertina, Mary, Frank and Charles.
Do you have anymore information about his parents and siblings? Without more to go on its not really possible to say if the above family is the right one.