Is it permissible for the mother to take half of the money designated for charity without the father's knowledge, given his shortcomings towards his family?

It is the wife's right that her husband spend on her according to what is customary. If he refuses, despite his ability to do so, then she may take from his money what suffices her and her children, without his knowledge, based on the agreed-upon Hadith of Hind. However, if he is spending according to what is customary, then it is not permissible for her to take anything from his money without his permission; otherwise, she would be sinful and betraying the trust.