There is little surviving information on J. A. Edwards.
Born (approx) in 1851, we know that he rode Millington in the 1878 Lincoln Handicap (unplaced behind 6-1 second favourite Kaleidoscope).
In that year a Mr J Edwards owned a 4-y-o colt (by Jove - Spec) which ran at Kempton on October 4: it was also ridden by a J Edwards, so it may have been a case of Edwards being the owner/jockey of the horse or, perhaps, they were two different people with, for example, the jockey riding for his trainer father (or brother).
The 1871 census shows a jockey of that name living at Woodcote End.