Here's an idea, but I don't know how well it would work. You could remove the other half of the cracked washer so that there is a uniform open space in the handle. Carefully clean out that opening and rough up the surface of the other washers with a strip of sandpaper. You could cut a slightly oversized replacement washer out of a fresh piece of leather of appropriate thickness. Then cut out the center of the new washer to accomodate the shaft of the hatchet. Make one cut in the washer that would permit you to slide it into the open space, over the shaft. Once you are confident that the fit isn't too bad, smear it with carpenter's glue and slide it in. After it has set, you could sand off the slightly protruding edges of the new, slightly oversized washer so it is smooth with the rest of the handle. The last step would be to sand the entire handle and varnish it.
If you are looking for a quick and easy fix that would include wrapping the handle with some sort of friction tape, you could remove the existing half-cracked washer and simply fill the space with a few wraps of leather thong that you could glue in. You could wrap with tape over that. It wouldn't be as pretty, but it would probably work OK.