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Looking at the photograph again, I believe this copy might have been printed back to front. The cap badge appears to show the Regiment Number in Roman Numerals, i.e., IIV. Reversed this would show VII, representing the number seven.

This would correspond to the Seventh Regiment of New York. The letter 'L' on the collar, I believe represents the Company, associated with the Regiment. I have seen some letter 'L' Company's described as Artillery.

https://www.ancestry.co.uk/search/collections/31388/

The above Ancestry Link describes the History of The New York 7th Regiment, and plenty more information can been gleaned from the W.W.W.

https://history.army.mil/html/museums/uniforms/survey_uwa.pdf

The above Link is for a PDF download of American Service Uniforms and Kit, throughout the ages.

I'm not too familiar with this Subject Matter, but the download gives valuable information on how Uniform Dress changed over the years. The Soldier depicted in the photograph appears to be Infantry, as the Cap Badge shows crossed Rifles. The top Coat appears to be the old style colour blue, which was gradually superceded by Khaki from 1902, though was continued to be worn by some Artillery and Engineering servicemen, until stocks ran out, approximately by 1911. As yet I cannot identify the Regimental Cap and Collar Badges.

Add: From information in the above Link, the Uniform Style could date from the mid 1890's. It was also adopted by the New York National Guard, begining around 1896.

Surprisingly, the GRO lists just one John Bailey, born between 1846 and 1850 Manchester, with no middle name.

BAILEY, JOHN       WELSH 
GRO Reference: 1847  S Quarter in MANCHESTER  Volume 20  Page 663

Quote from: Bee on Tuesday 22 June 21 23:52 BST (UK)

OH's 3xgt grandfather married and had 5 children, first wife dies, he then marries the illegitimate daughter of his first wife and had a further 8 children 

Was that legal? 

The illegitimate daughter would be His stepdaughter. According to the link below and assuming this was in the 19th Century, the answer is, that it was not legal.

https://books.google.co.uk/books?id=5NiHAgAAQBAJ&pg=PA12&lpg=PA12&dq=in+the+19th+Century+could+a+man+marry+his+stepdaughter&source=bl&ots=VDG74PHCBA&sig=ACfU3U1ifEu5a14Z4FzG_aD5vvmt7sTmCg&hl=en&sa=X&ved=2ahUKEwiO85r29uDyAhWYh_0HHSGOBVYQ6AF6BAg1EAI#v=onepage&q=in%20the%2019th%20Century%20could%20a%20man%20marry%20his%20stepdaughter&f=false

https://digitalcollections.nypl.org/collections/thomas-a-larcom-photographs-collection#/?tab=navigation&roots=1:927c8ef0-c6bd-012f-1ca3-58d385a7bc34

Above is the Link, to an Online Collection of Mountjoy Criminal Photographs.