is there a tikkun for karet that one learns thursday night? what are the details?
I don’t know of such a procedure. However, the Rambam writes that there is nothing that stands in the face of teshuvah, even sins that involve karet, and sincere teshuvah, including of course entirely leaving the sin, confession, and acceptance for the future, can work for anything.
Having said this, there are many levels and many ingredients in full repentance, as we find in the sefer Sha’arei Teshuvah, and the great leaders of many generations also gave advice of which practices can be adopted to assist in reaching full atonement.