If it's the (William) Arthur Pay mentioned by Rosie in NY in 1910 (born c 1864 rather than 1854), is this his birth ref?
PAY, WILLIAM ARTHUR Mother's Maiden Name SMITH
GRO Reference: 1865 D Quarter in
SAINT GEORGE HANOVER SQUARE
Volume 01A Page 283
Whoops - Thanks for correcting me Igorstrav

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Is it me as I can't see a marriage for Ada Smith and WJP.

I am wondering if she was actually married to WJP

I am just posting this as I probably won't find it again

She gives her age/place of birth as c1844 Holborn on census.
1851 - HO107/1467 f485 p28
George Pay 34 occ Coachman bn Huntington, Sussex
Mary Pay age 42 bn Marylebone
George Pay age 14 bn Islington
Ellen Pay age 11 bn Whitechapel
Ada Emily Pay age 6 bn HolbornEdward
Smith (Nephew) age 3 bn Lambeth.
(George Pay married Mary Smith
30 Jul 1836 at Old Church,Saint Pancras,London)
In 1861 Ada E is possibly a servant
RG09/87 f79 p21
........
When she married in 1885 she named her father as George Smith.
She could be pretending to be married, she would not be the first to do that
ADDED
I see that when she married in 1885 her father George Smith's occupation was a Coachman, the same as George Pay in 1851