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Looking again I feel that it is around Whanganui.  It may not be on the RHS going from Sanson but the LHS has pine trees in the light soiled dune country to the west.  The second idea is that it is on one of the back roads east  or north of Whanganui.  Even accessing the Parapara.

Trish18 with her fantastic knowledge strikes again, in the best way possible.....and with more photgraphs.

Thanks Trish18 for your work.  It is appreciated. I'm in NZ and am always pleased when the sleuths have a successful 'sleuth'.

If William Arthur's names are not from his parents etc it may be that the Arthur is from the Duke Of Wellington. A serving soldier would be aware of his commander.  The DoW was reputed to have been born in Dublin and perhaps James Bennie/nney was also from there. 

https://en.wikipedia.org/wiki/Arthur_Wellesley,_1st_Duke_of_Wellington

There are Berney/Beirney families from Ireland.

Thanks, Bowen was what I thought.

@Tickettyboo
It was a birth certificate of Mary's daughter in 1887.
Mary was married to William Matthews. They had their first child in either 1883 or 1884.
Mary was born approx 1863, possibly in Ireland. I've not found a UK marriage so maybe they were married in Ireland.

If you would like us to search further then it would be good to have further info from you. 

In the UK many families named their children according to a pattern.  In Ireland this is the naming pattern. Not saying that the Irish  would slavishly adopt this but it can give clues.

From Mrs Google

'Traditional Irish naming conventions dictate that the first son is named after the father's father, the second son after the mother's father, and the third son after the father. For daughters, the first daughter is named after the mother's mother, the second daughter after the father's mother, and the third daughter after the mother. These patterns could be used to help trace family history, but parents might also use nicknames or name a child after a deceased sibling.
 
Typical Pattern for Children
First Son: Paternal grandfather (father's father)
Second Son: Maternal grandfather (mother's father)
Third Son: Father
First Daughter: Maternal grandmother (mother's mother)
Second Daughter: Paternal grandmother (father's mother)
Third Daughter: Mother'

So if you have the names of the children of the Matthews/Bowen marriage we might be able to help.

If you could list them in date order that would be great, pl include the names of any children who died.