If the woman you mention is from one of the groups mentioned in the 60th verse of the Surah At-Tawbah, you can give your Zakat to her. You don’t need to tell her that it’s your Zakat.
 “Zakat expenditures are for the poor, the desperate, those employed to collect Zakat and those to warm their hearts (to Islam); and are used for freeing captives, for those in debt, for the cause of public interest and for the people that work in public service. This is an obligation by Allah. Allah is All-Knowing and The Wise.” (At-Tawbah/The Repentance 9:60)