In the very beginning of the movie, when Nick's fighter is counted out, in the close up, Nick puts his cigar in the right side of his mouth. Then, in the very next shot, a bit farther away, the cigar is in the left side of his mouth.
At about the 45 minute mark into the film and after the "first" professional fight. Galahad knocks out the more intimidating opponent. The ref's count was extremely fast...but as soon as he counted "...ten" and waved his hands signifying the bout was over, the defeated opponent was hurriedly getting up after being "knocked out"
Kid Galahad trains for the climactic big fight during the middle of summer, Independence Day to Labor Day, yet when you see all the orange and yellow leaves on the trees and on the ground it is obvious that these scenes were filmed in autumn.