Is the prayer valid when repeating the words of Al-Fatiha that were mispronounced until corrected, or is it a condition that the recitation be correct from the first time?

Whoever makes a mistake in Al-Fatihah that invalidates the prayer must re-recite what they erred in, and their prayer is not invalidated by that. The letters of Al-Fatihah must be observed; if one omits a letter from it, they must re-recite the word containing that letter and what comes after it, provided there isn't a long interval or bowing, otherwise the prayer is invalidated. One should beware of waswasah (whispers/obsessive thoughts) regarding the recitation of Al-Fatihah.