Although this doesn't appear to match with any of the known silent films titled "Sidewalks of New York" - the title contained in the decorative lozenge in the border of the intertitle - I believe that this *is* very likely the 1923 Lester Park Productions film of that title.
For one thing, the actor seen in close-up in Still 5/5 is Bernard Siegel, who appears in the credited cast of that film. (Siegel later appeared prominently opposite Lon Chaney in "Laugh, Clown, Laugh").
The AFI Catalog describes the 1923 "Sidewalks of New York" as a prizefighting drama revolving around a young woman's exploits in the Ladies' World Championship.
However, contemporary newspaper ads for the film from 1923/24 refer to its star as "Little Hanna Lee, the talented child actress", and a brief capsule write-up of the movie in the Fitchburg Sentinel (Massachusetts), August 15th, 1924, p.5, reads as follows: "A modern Cinderella, a little waif held captive in the den of a brutal bootlegger, is the personage about whom the story revolves, the part being wonderfully played by Hanna Lee, a child actress of remarkable talent."
Robert Kiss, 21.09.2010