May a woman who has lice cut her hair during the nine days to eliminate a problem with chatzizah.
Yes, this is fine.
The question of a haircut for women in the Three Weeks is disputed among authorities.
Some maintain that it is permitted for women to take haircuts even during the week of Tisha Be’Av (see Panim Meiros, vol. 2, no. 37), and this is the custom for Sephardic women (see Chazon Ovadyah, Taanios p. 163; see Or Le-Tzion vol. 3, chap. 27, no. 3, who writes that although it is permitted for women to take haircuts throughout the Three Weeks, it is preferable to defer the haircut until after Tishah Be’Av).
For Ashkenazim, some authorities rule that women must be stringent with regard to haircuts from Rosh Chodesh Av (Noda Biyehudah YD 99; Pri Megadim, Mishbetzos Zahav 551:13; Kaf Hachaim 551:47). The Mishnah Berurah (551:79; see also Mishbetzos Zahav 551:3, and Derech Hachaim p. 169) writes that there is no distinction between women and men, and haircuts are prohibited for all (Ashkenazim).
In principle, it is therefore right for Ashkenazi women to act stringently In this matter. However, the Mishnah Berurah (ibid.) writes that for hair growing on temples a woman may be lenient, and Halichos Shlomo (p. 414) states that the same applies to other forms of cosmetic treatments (eyebrows).
The same ruling will apply for a haircut that is necessary to take care of lice. Just as a woman is permitted to beautify herself, even by means of cutting hair, so she is permitted to do all that it required for the taharah procedure, even cutting hair (to eliminate lice).