On the marriage certificate, what were the names of the witnesses?
I note that the 1872 birth certificate has under the same reference both Herbert Amot and Herbert Atkinson listed.
In Barnsley, the surname Amot/Amott:
Herbert Amot, b. Dec 1872 Barnsley (aka Herbert Atkinson?)
Thomas Amott m. possibly Jane Atkinson, Mar 1874 Barnsley
Hannah Amot, b. 1874
Mary Ellen Amott, b. 1876
Thomas Amott, b. 1879
They then seem to have ended up in Doncaster district where marriages for Hannah, Mary Ellen, and Tom are found under the Amot(t) name.
In 1881 they are mistranscribed as Arnott and living in Worsbrough. There are both Atkinson and Arnott(Amott) children in the household; it appears that Jane was older, had been previously married, and Herbert may have been born after her first husband's death but before she married Thomas.
In 1871 Jane and children are with her husband Benjamin Atkinson. He died Sep 1871 quarter; i.e. he could not have been Herbert's father.
Benjamin Atkinson m. Jane Rodgers in 1862; in 1871 she also had with her a 13 year old son Fred Rodgers.
In 1861 they are listed as living together as husband and wife - several other Atkinson children (probably from another marriage of Benjamin's?), and "Alfred" Rodgers aged 2 are in the household, as is a son of Benjamin and Jane's, Paul.
So it appears that Jane either had a marriage before Benjamin or had an illegitimate child Alfred/Fred in about 1859, was living with Benjamin by 1861 (son Paul registered as Paul Atkinson Rodgers), married him the next year, he dies in 1871 and she soon is with Thomas Amott (the most likely candidate for Herbert's real father), then marries him and has a few more kids (they are together up until the 1911 census!).