If one is accustomed to daven mincha until 58 or so minutes after shkia, based on the opinion of Rabbeinu Tam and the Shulchan Aruch, and as is the custom of certain Chassidic Jews, is it permitted to light Chanukah candles first (at shkia, or 10-20 minutes after shkia) and then daven Mincha 30 minutes after shkia)? Or is it a contradiction? If one did light candles and then reminded himself that he did not daven Mincha, is it forbidden for him to daven Mincha now?
Technically it is a contradiction and preferably it shouldn’t be done, however if he did light, he may still daven mincha. This is the way that it is done on erev shabbos, that many people light chanukah candles and then daven mincha. As a side point, it isn’t possible to light the candles all the time on the “right” day, and on and on erev shabbos they can only be lit when it is the “wrong” day, and before one accepts Shabbos.
However it is always better to daven mincha before lighting Chanukah candles. One of the reasons given is because mincha is to represent the korban tamid, which always came before the candles in the bais hamikdash.
O:CH 679-1, Pri Megadim Eshel Avrohom 671-10, Birkei Yosef 679-2, Ben Ish Chai Vayeshev -20, Yalkut Yosef pg. 212
As an interesting point see Halichos Shlomo om chanukah pg. 317 regarding if someone lit the first candle before shkia and then davened mincha if he should say al hanisim.