Is giving a portion of money designated for charity to needy relatives (a paternal aunt's husband's wife and a maternal aunt) considered a betrayal of trust?

If the woman has given your mother an absolute power of attorney to distribute the money, and she has not specified a particular person, then there is no harm for your mother in distributing it according to her discretion and what she deems appropriate. It is permissible for her to give it to those you mentioned, given their difficult circumstances.