When I looked up Damon Runyon on Wikipedia before writing this comment, I could find no reference to this story or this film. This has to be the most obscure Damon Runyon story ever put on film.
As surely as Mark Twain is accepted as the chronicler of life on the Mississippi, Damon Runyon is his counterpart for life on the most famous street of all, Broadway in New York City. His characters whether he writes seriously or for comedy are a part of New York, they could not exist in any other environment.
Johnny One-Eye is one of his serious and more obscure works. Old time gangster Pat O'Brien and Wayne Morris have done their share of bad things, but now have risen to some respectability. But when a hot shot prosecutor starts breathing down Morris's neck, he's ready to feed him O'Brien. That O'Brien won't have and when he confronts Morris, he's forced to shoot and kill one of Morris's gunsills. O'Brien is wounded himself.
The wounded gangster is befriended by a little girl and her badly injured dog who has only one good eye. O'Brien has quite an interesting 24 hours before all is resolved.
The film it bares the most resemblance to that I've seen is Odd Man Out. O'Brien certainly found a few more friends than James Mason did in his injured state though.
Sad to say that this film is a cheaply made independent film with not so great production values. That's made up by the location shooting in New York a la Naked City. Look for a good performance by Dolores Moran as the showgirl mother of the little girl befriending O'Brien.
Fans of Damon Runyon might want to give this one a look.