I think the John Morgan who printed this sheet must be the ballad writer of that name profiled in Charles Hindley's History of the Catnach Press (1886). Here's an extract with Hindley describing his attempt to interview Morgan for the book:

“We talked and took notes. Mr Morgan talked and took gin. Mr Morgan got warm - warmer and warmer - and very entertaining. His conversational powers increased wonderfully. He became very witty and laughed. He joked and made merry at some old reminiscences in connection with old Jemmy Catnach, and admitted that, after all, old Jemmy wasn't a particularly bad sort. That is, when you knew him and could handle him properly. Then old Jemmy was as right as my leg! Still we continued to talk and take notes. Still Mr Morgan talked and took gin. [...] At length, it became very manifest that we should not be able to get any more information out of Mr John Morgan on that day.”

Morgan was one of the 19th century ballad trade's best-known, most talented and most prolific writers. My guess is that he later branched out into printing as well to avoid splitting the proceeds of his work with an outside printer like Catnach.

I've written a little more about both men in my British Broadsides article here. Scroll down to the appendices and you'll find some details of another delightful suggestive ballad sheet called The Beautiful Muff.