Is it permissible for a poor, indebted employee to take what remains of the zakat money he was entrusted with distributing, due to his poverty and need?

If the one who entrusted you with the Zakat has given you a general authorization to distribute it, and you are poor and in need, then it is permissible for you to take from it. However, if the one who entrusted you specified particular individuals or entities, then it is obligatory to pay it to those whom they specified, otherwise you will be held liable for the money.