Arthur George Skelton
Sep 1896 Caistor 7a 626
(Caistor district covered Great Grimsby until 1897)
Probably the only way to are going to possible find who the father is of Arthur is to obtain his birth or marriage certificate - hopefully mother Lucy has named the father.
Difficult to say whether the birth entry above is your Arthur it does look promising. As Arthur's middle name is George that might indicate that George Skelton is his father.
On your Lancashire post 'Help on tracing Peter Sanderson b 1832 Carleton' you mention that Gertrude (father James Penswick mother Margaret Sanderson) marries Arthur George Wilkins
Do you have access to this marriage certificate?
If not an option would be to order this and see who Arthur has named as his father.
Unfortunately, as far as I can see Lancashire OPC do not have the entry - St. Chad's Poulton Le Flyde marriages only go up to 1900 - you would need to purchase the certificate unless someone can help with a look up at the record office in Preston, Lancashire
Here are the details from Lancashire BMD
PENSWICK Gertrude/WILKINS Arthur George St. Chad Poulton-le-Fylde, Registers at Preston reference number FJ/3/203
Here are the details from FreeBMD
Dec qtr. 1919 Flyde Vol 8e page 1777
Arthur George Wilkins
Gertrude Penswick
Here's a picture of the church from Lancashire OPC
http://www.lan-opc.org.uk/Poulton-le-Fylde/index.html
This is where it gets most interesting. For whatever reason, Lucy leaves James Wilkins and moves to Grimsby.
The notes from Amy says she was a housekeeper with a man named "Barker"
This is the first listing in the census of my grandfather w/ Lucy, but with a man named "Skelton".
1901 RG13 Piece:3091 Folio:186 Page:36
SKELTON, Geo Head Married M 38 1863 Fisherman Aldford Lincolnshire
SKELTON, Lucy Wife Married F 38 1863 Rochford Essex
WILKINS, Ethel M Step Daughter Single F 16 1885 Waitress Aldershot SKELTON, Sarah A Daughter F 15 1886 Grimsby Lincolnshire
WILKINS, David J Step Son M 14 1887 Nipper Lad Railway Westminster
WILKINS, Amy E S Step Daughter F 12 1889 Westminster London
SKELTON, Arthur G Son M 4 1897 Grimsby Lincolnshire
As Arthur was born after 1891 census and before 1901 census difficult to say whether Lucy was expecting before she's on the next census (1901) with George Skelton.
As no marriage found Skelton/Wilkins perhaps she meet George while expecting (maybe by father named Barker ) and he took her and the Wilkins children in and when Arthur was born she gave him his name.
When she left George Skelton for what ever reason after 1901 and meets Stephen Hazelgrove then marries him 1912. She gave daughter Phyllis Elizabeth Hazelgrove surname of Wilkins (1906) which would seem to indicate Stephen Hazelgrove to be her father. Lucy's maiden name was RILEY - Lucy Ellen Riley married James Wilkins in 1883.
Before 1911 census perhaps after birth of Phyllis 1906 Lucy then changed her and Arthur's surname from Skelton back to Wilkins because she was not married to George Skelton only living with him sometime after 1891 census.
The only way I can see to unravel your mystery is with Arthur's birth record and hope that Lucy has named the father.
I would assume on his marriage certificate he is likely to name the person known to him
