Is it permissible to allocate an amount from the annual Zakat (128 Egyptian pounds) for building a mosque, in response to a request from someone who helped find accommodation, and is it permissible to divide this amount into three monthly installments, especially given the existence of debt and high living costs?

It is not permissible to give zakat money for building a mosque. You are not obliged to fulfill what this person asked of you unless it was a fee for helping him; otherwise, this charity is not binding on you. If you wish to fulfill his request, you may do so, but do not consider what you paid as zakat. It is not permissible to give a single amount of money for both zakat and this matter. It is permissible to pay zakat in installments before the due date (hawl), but it is not permissible to delay it after the due date. As for debt, its ruling and the scholarly اختلاف (difference of opinion) regarding its deduction from money subject to zakat have been explained in Fatwa No. 100534.