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I stumbled on this query today while researching a DNA match to a Dorman family that shares your descent from the subject of your query.

Here's what I found:

John James Sowerby Carter, bachelor, son of John Sowerby Carter, married, at St. Mary's in Birkenhead, 29 Jan. 1859, Catherine Mary Blain, widow, daughter of Joseph Garner:

https://familysearch.org/ark:/61903/1:1:F3FM-94Y

In the 1851 census, John James Sowerby, a bookkeeper, 22, born Liverpool, was enumerated on Woodchurch Road, Oxton Birkenhead, in the household of the bookkeeper Thomas Nichols:

https://www.familysearch.org/ark:/61903/1:1:SG29-N3L

This is presumably within a mile or so of where your John and Catherine Carter were living in the 1861 census:

https://www.familysearch.org/ark:/61903/1:1:M7GC-X5F

John James Sowerby, son of John and Emma, was baptized at St. Peter's in Liverpool on 12 Dec. 1830:

https://familysearch.org/ark:/61903/1:1:NVYV-457

The OnLine Parish Clerk transcription of the baptism has the father, John Sowerby, as a mariner, abode Whitechapel:

http://www.lan-opc.org.uk/Liverpool/Liverpool-Central/stpeter/baptisms_1830s.html

John Sowerby married Emma Thornburgh in 1823 at the Church of St. James in Liverpool:

https://familysearch.org/ark:/61903/1:1:NX4C-6QT

The marriage intention, dated 25 Aug. 1823, is indexed here, with the prospective groom's name mistranscribed as "Lowerby": 

https://familysearch.org/ark:/61903/1:1:FBX7-7R4

Other children of John Sowerby and Emma Thornburgh:

Emma Adelaide Sowerby, of 42 Whitechapel, baptized at St. Peter's in Liverpool on 19 Dec. 1831:

https://familysearch.org/ark:/61903/1:1:NNHZ-C2K

Francis Henry Sowerby, baptized at St. Peter's on 15 Apr. 1834:

https://familysearch.org/ark:/61903/1:1:NYSP-33P

The OnLine Parish Clerks transcription of this entry records the father John Sowerby as a master mariner, of Sefton Street, Toxteth Park:

http://www.lan-opc.org.uk/Liverpool/Liverpool-Central/stpeter/baptisms_1834i.html

and the following St. James, Toxteth, burial entry indicates that Francis died aged 9 months, and was buried on 20 Aug. 1834:

http://www.lan-opc.org.uk/Liverpool/Toxteth/stjames/burials_1827-1835.html

Emma Sowerby, widow, of Toxteth Park, married next, 7 Jul. 1835, at St. Mary's, Walton-on-the-Hill, Lancs., William Carter, teacher, Toxteth Park.  The witnesses were John Bell and James Richmond.  In both the OnLine Parish Clerks transcription and the various Family Search transcriptions, the surname Sowerby is incorrectly transcribed as Lowerby:

http://www.lan-opc.org.uk/Liverpool/Walton/stmary/marriages_1831-1837.html

(Married the same day at St. Mary's, John Bell, Millwright, and Jane McCarter, spinster, witnesses Jane Robinson and William Carter.)

William and Emma  Carter were enumerated in 1841 at Bidston, Cheshire, with their daughter Sarah Carter, 4, and Emma's two children, John Sowerby, 10, and Emma Sowerby, 9:

https://familysearch.org/ark:/61903/1:1:M7M1-C35

Sarah Ann Carter, daughter of William and Emma, was baptized at St. Peter's in Liverpool on 26 Mar. 1837:

https://familysearch.org/ark:/61903/1:1:J9M8-4X8

Matilda Carter, daughter of William and Emma of London Road, her father a school master, was born on 6 Jan. and baptized at Christ Church, Hunter Street, Liverpool, on 10 Jan. 1838:

http://www.lan-opc.org.uk/Liverpool/Liverpool-Central/christchurchh/baptisms_1826-1838.html


Good luck with your research --

John