Some people recite Tehillim chapter 91 during this half-hour. The Sefer Zechirah (69:1) writes, “A segulah that helps to be protected the entire year is to say seven times (Tehillim 91) עליון בסתר יושב, each night of Chanukah after lighting the candles." Some say chapter ט"קי in Tehillim because it has many requests for success in Torah and mitzvos.
Whenever one performs a mitzvah, it is a good time for tefillah. As is brought in Tosefta (Maasar Sheni 5:15), “When one performs a mitzvah, it is an ideal time to daven to Hashem.” This is particularly true after lighting the lecht of a mitzvah (Such as Shabbos or Chanukah lecht). Rabbeinu b’Chaya (19:3) writes, “When a woman lights the Shabbos candles, she should daven that she merit children talmidei chachamim, who will shine with Torah because tefillos are more accepted after performing a mitzvah."
The Shabbos and Chanukah lecht are mesugal for children talmidei chachamim, so it is good to daven for that at this auspicious time.
The Great Test: The Beis Avraham (רק ה"ד 'ד נר) says that אלא בלבד לראותם implies that looking at Chanukah lecht is a refuah and rectification for sins related to guarding the eyes.
