Salat-ul-Khusuf as referred to in the question is valid. However, making Du`a' during it as mentioned in the question is not authentically reported from the Prophet (peace be upon him) or from the Rightly-Guided Caliphs (may Allah be pleased with them all). Consequently, such Du`a' is Bid`ah (innovation in religion) for the Prophet (peace be upon him) said:
He who innovates things in our affairs for which there is no valid (reason) (commits sin) and these are to be rejected.
but it does not nullify the Salah.May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.