How authentic is the ḥadīth reported concerning the virtue of prayer on the last Friday of Ramadan, which states performing four rakʿahs with one tashahhud, reciting Al-Fātiḥah, Al-Kawthar, and Al-Qadr 15 times in each rakʿah, with the intention of atoning for missed prayers, and that it expiates for 400 or 1000 years, attributed to Imam Ali?

Whoever misses a prayer until its time has expired due to an excuse, such as sleep or forgetfulness, must make it up immediately. As for whoever misses it intentionally, they must repent and seek forgiveness.

What is narrated about a prayer performed on the last Friday of Ramadan to make up for missed prayers (known as "Salat al-Qada al-Umri" or "lifetime make-up prayer") is a lie and an innovation that has no basis in Islamic law. It was not known to the Prophet (peace be upon him), nor to his Companions, nor to the imams of guidance. The Prophet (peace be upon him) clarified that making up for missed prayers is for one who slept through it or forgot it, by praying it immediately upon remembering it, not at a specific time.