What is the expiation for one who prayed knowing there was a possibility of a portion of his back becoming uncovered during prostration, then became certain of its uncovering after the prayer and repeated it? And is his act considered like that of one who prayed intentionally without ablution?

The back of a man is not part of the awrah (private parts). So, if any part of it that is parallel to or above the navel becomes uncovered during prayer, it does not affect the validity of the prayer. This is because, according to the most preponderant opinion of the scholars, a man's awrah is what is between the navel and the knee. If a part of the awrah becomes uncovered, then covering the awrah is a condition for the validity of prayer, and it is not permissible to enter into ihram (the state of ritual consecration) while being uncertain about fulfilling a condition for the validity of prayer. If he enters ihram and completes his prayer in that state, his prayer is not valid, and he must repeat it. There is no expiation for what he did other than repeating it and repenting if he was negligent in his prayer. A slight uncovering of the awrah during prayer is excused by the Hanbalis if it occurs unintentionally.