I'm very sorry to hear that.
As soon as the chicks reach weaning age, it's recommended to remove the mother budgie in order to prevent aggression from happening (as well as a double clutch).
At this point and given the injuries your chick has sustained, the best course of action is to remove the mother and to let the father take full charge of both chicks and finish raising them.
Due to this hen's more violent temperament, it's not advisable to let her breed again.
If this current pair of yours and the chicks are in an aviary/colony setting and there are other budgies sharing the same enclosure, then that will add further problems, because the chicks can still be attacked, especially by other hens and in this case, moving the father and the chicks into a cage would be the safest procedure. You will have to keep a close eye on the chicks and if they aren't being well taken care of, then you will have to give them supplement hand feedings and to encourage them to try out all the different types of food.
From your description, it's possible the chick has hatched with a beak deformity and as long as the little one is able to eat well, then this shouldn't be a problem.