1. Make it more religious. You could keep most of the show intact and have it be believable if it were more of a religious parable, morality play, etc. You know who's good and who's evil. You're waiting for the bad people to get punished and the good people to triumph. Just add in more lessons about hard work and prayer and belief in God and we'll be rooting for the family to be reunited instead of wondering if the show really doesn't realize we've already figured it out. The bones are already there with her family and the quasi gospel/church song near the end. You could also abandon whatever subtlety they were trying for with the mayor which makes him rather bland. By adding in more lessons about ambition, hard work, etc. it might also (barely) make the mother and son seem more similar and both obscure that secret and have it seem less suspicious that she keeps helping him and handing opportunies to him.
2. Fix the scene where they throw the baby from the train. Not only is it very predictable that the baby will live when you see the bag flying slowly and dramatically overhead but it's stupid. It's ridiculous. There has to be a better way to stage that. If nothing else, just have him throw it through the window upstage.