What is the authenticity of the story of the barren woman for whom Moses, peace be upon him, prayed, and she conceived, and when Moses asked his Lord about that, He answered: "Every time I decreed her barren, she said: 'O Most Merciful,' so My mercy preceded My power"?

There is no basis for the story mentioned; it is among the Israiliyyat (Judaic traditions) that are not trustworthy. As for divine decree (Qadar), Allah knows everything that will be and it is written in the Preserved Tablet (Al-Lawh Al-Mahfuz), and this does not change. There are other writings by the hands of angels, such as the writing of a person's destiny while they are a fetus, and these can change due to causes, such as maintaining kinship ties or supplication (Du'a). For Allah erases what He wills and establishes what He wills, and the Mother of the Book (Umm al-Kitab) remains with Him, unchanged.