One should not recite the Havdalah over the Chanukka candles since the blessing of "...Who creates the lights of fire" [one of the three Havdalah blessings] requires one to have some benefit from the light and these candles are forbidden to benefit from.
However, it is permitted to first recite Havdalah, extinguish the candles used for Havdalah and then use those same candles for the Hanukkah lights (Bach). In fact, this is a preferable practice, for since you have performed one Mitzvah with them let another be performed with them as well (see Taz).
Of course, whether this practice should be followed first depends on what we have yet to resolve: Is Havdalah or Hanukkah candles first?
Our teacher the Beis Yosef writes:
- "The Chanukka candles are lit in the synagogue before Havdalah is recited."
Until here is the quote, whose reason is explained in his great work [The Beis Yosef commentary on the Tur]: Because we are trying to delay the new incoming day [we are trying to prolong the Shabbos as long as possible], (see there). Now if this is so, why specifically in the Synagogue? The Rama indeed writes on this and says:
- "All the more so in one's home, as they have already heard Havdalah in the Synagogue (Shul)."
So are his words. We learn from here an implication: Since one has already heard Havdalah in Shul he may light first, however normally Havdalah would be first. This would seem to be the law, since it is difficult to conceive of the blessing "Who created the lights of fire" (Borei M'orei HaEsh) being recited after one has already lit fire for the Menorah. This reasoning is also recorded in the Kol Bo in the name of the Raavad of blessed memory. Even though it could be argued that there is no benefit derived from the Menorah there is still the use of fire [before the blessing that recognizes its permit]. Furthermore, one can at least benefit from the service candle (Shamash). Therefore the law is to recite Havdalah first in the home. It is only in the synagogue, with its greater ability to publicize the miracle, where it is deemed to be preferable to light the Menorah first. This then would also be true in the home for one who already listened to the Havdalah in the synagogue. However, lacking this, the Havdalah comes first. This is our current practice.
Note further: The main reason given, that of "prolonging the Shabbos" - is really not a satisfactory reason in and of itself. This is because it would only apply if we continued to observe the laws of Shabbos. Since we are lighting the Menorah the day of Shabbos must certainly be over. In addition, Havdalah should have the status of the more common occurrence (Tadir), and so have precedence in law, as well as being more appropriate for the Shabbos' exit. The Chanukka candles really belong to the following day. Even given all these points, this practice, which is limited to the synagogue, is kept for the reasons we have previously stated.
(The 'Magen Avraham' writes that one who forgot to say "You have graced us..." ('Ata Chonantonu') [in the evening prayer] should recite Havdalah first (see there). It is therefore implicit that he is of the opinion that each individual is usually lighting the Chanukka candles first. The 'Eliya Rabbah' has written so as well, and this is the implication of the words of the 'Gra'. However, the 'Taz' and 'Maharal' from Prague rule to recite Havdalah first. This later opinion appears to be the primary ruling on the matter, in my humble opinion, for the reasons that we have written and because the former opinions [are forced by their ruling to] reject the words of the 'Taz'. Regardless, according to what I have written the issue seems dealt with appropriately. Attend to the sources and you will see it is so.)