Is it permissable to have a hair cut on 10th Tevet (according to Halocha lemaasa, Kabolah and Reb Yehudah HaChassid)
There is no formal prohibition on taking a haircut on the Tenth on Teves.
However, some write that a haircut should preferably be avoided. The source for this is the Gemara in Rosh Hashanah (18), where Rabbi Yehoshua took a haircut to show his disapproval of the enactment of a fast day on Chanukah.
This indicates that it is (at least) improper to take a haircut on a fast day, for otherwise why should the haircut be an expression of disapproval. Based on this, the Ruach Chaim (Siman 566) writes that it is not right to take a haircut on fast days.
The Turei Even, however, points out that we do not find a restriction on haircuts for fast days other than Tisha Be’Av.
It is interesting to note that the Mishnah Berurah (550:6) writes that a person who can (and is a ba’al nefesh) should preferably accept the stringencies of Tisha Be’Av even for the other fasts. This appears odd: Are we supposed to avoid washing our hands on all fast days?
It is possible, however, that the intention is to act in accardance with the halachos of the nine days–meaning that it is better not to take a haircut.