A:
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.
(Related by
Al-Bukhari and
Muslim),
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.