What is the appropriate time of night for prayers to be answered, and has it been reported from the Prophet, may Allah’s prayers and peace be upon him, that there is an hour of the night during which prayers are answered without fail?

At night, there is an hour of a vague response every night. The Messenger of Allah, peace and blessings be upon him, said: "Indeed, there is an hour in the night in which no Muslim man supplicates to Allah for good, concerning the affairs of this world or the Hereafter, but Allah grants it to him, and that is every night." The most hoped-for time for this hour is the last third of the night.