Am I obligated to inform my mother that I am sponsoring an orphan girl from my own money, which I receive from her as an allowance, knowing that I fear hypocrisy for myself?

If the money was given as a gift (for ownership), then the daughter is not obligated to inform her mother what she did with it. However, if it was given as anadesignation for her own sustenance, then it is necessary to seek the mother's permission to sponsor the girl, because the remaining money does not belong to the daughter; rather, it belongs to the mother. Therefore, it is not permissible to dispose of it in a way other than what the mother specified.