Musannaf Ibn Abi Shaybah

Hasan was asked: If a man misses one rak‘ah, joins the prayer, then the imam’s wudu’ breaks, so the imam takes the hand of this follower and puts him forward, what should he do? Hasan replied: He should pray one rak‘ah and sit down, then build upon the prayer of the people. When he has led them in four rak‘ahs, he should sit and recite the tashahhud, then take the hand of another man and put him forward. When that person says salam, he should stand up and complete his missed rak‘ah.

Ibrahim was asked about a person who was appointed as the deputy by the imam after missing one rak'ah, so what should he do? He replied that he should lead the people in the remaining prayer, and when he completes the prayer, he should bring forward a person who has performed that rak'ah; when he finishes with salam, then this person should stand up and complete his prayer.