Is it legislated to supplicate with the supplication of the prostration of recitation (sujūd al-tilāwah): “O Allah, record for me thereby a reward with You, and remove from me thereby a burden, and make it for me a treasure with You, and accept it from me as You accepted it from Your servant Dawud” in every prostration other than the prostration of recitation?

There is no objection to supplicating with the supplication of the prostration of recitation and persisting in it during prostration, whether it is an obligatory or supererogatory prayer, even if it is for something other than recitation, due to the generality of the Prophet's (peace and blessings of Allah be upon him) saying: "The closest a servant is to his Lord is while he is prostrating, so increase in supplication," and his saying: "As for bowing, glorify the Lord Almighty therein, and as for prostrating, strive in supplication, for it is likely to be answered for you." The best supplications are those that are transmitted (from the Prophet), and among them is the aforementioned supplication. Khalil al-Maliki said: "And supplicate with whatever you love, even for worldly matters," and Urwah ibn al-Zubair said: "Indeed, I supplicate to Allah for all my needs in prayer, even for salt."