Is it permissible to give monthly charity to orphans who live with their working mother, or should it be directed to a mosque under construction?

Voluntary charity (Sadaqah) is permissible for orphans, whether they are rich or poor. As for Zakat, it is not permissible for orphans unless they are poor, because Allah has specified the recipients of Zakat to be the eight categories mentioned in Surat At-Tawbah, in His saying, the Almighty: (Indeed, [prescribed] charities are only for the poor and the needy, and those who work on [collecting] them, and those whose hearts are to be reconciled, and for freeing captives, and for those in debt, and for the cause of Allah, and for the wayfarer - an obligation [imposed] by Allah. And Allah is All-Knowing, All-Wise).