The mule represents manhood because manhood often symbolizes a crossroads in life, and making the right decision "makes you a man". Jenny, who Dave has shot, represents that crossroads for Dave. It presents him with a variety of different options. After he shoots her, he can lie about it, try to cover it up, or confess to it and take responsibility for his actions. As children, we don't have to take responsibility like an adult does, so the decision that will confirm Dave's status as an adult would be to confess and take responsibility. However, Dave fails this test of manhood. He lies. He whines. He rants and rails internally. And in the end, he commits the ultimate act that seals his cowardice and lack of responsibility: He skips town, bailing out on all of his responsibilities.
Jenny represented to opportunity for Dave to prove his right to be called a man. Unfortunately, he behaves more like a petulant child, and affirms the second half of the title of the story, indicating that he was "Almost a Man". He wasn't there yet, and Jenny brought that to light. I hope that helps!