D..
First, I'd say point blank to place B's supervisor that you need more training and will not be able to finish something that has been outstanding for a month without proper training. Don't be apologetic. Don't be rude. Be matter-of-fact. If he says he cannot train you, thank him for being upfront with you, and then go to the person who hired you and ask for someone who has the time to train you since the supervisor does not. Tell this person that you have inherited "X". Don't call it a mess. Give it a name. Tell him or her that it has been outstanding for a month and without proper training, you will not be able to complete it in a day. You need better training. Ask how to get it.
And say nothing about scapegoats.
You should also tell the person who hired you that things are going well with place A's work and training and that you expect to be as successful at place B once the kinks are ironed out.
And as the other ladies say - document, document, document.
Dawn