Is it permissible to supplicate for anything, such as asking for concealment from disgrace after repenting from a grave sin, or is supplication restricted to specific matters?

It is permissible to supplicate with any wording, and it is not restricted to what is mentioned in the Quran and Sunnah, due to the saying of the Prophet, peace be upon him: "Let one of you ask his Lord for all his needs, even if it is for the shoelace of his sandal if it breaks," and his saying: "Then let him choose whatever supplication he wishes." So there is no harm in saying: "O Allah, do not expose me." However, it is preferable to adhere to the Quranic and prophetic supplications, such as: "O Allah, forgive me," for forgiveness includes concealment and overlooking.