Is the prayer of one who joined the congregational prayer late, then was overcome by forgetfulness and counted the rak'ah, then remembered it was not a valid rak'ah, correct?

If the latecomer (masbuq) does not catch the bowing (ruku') with the Imam, then that rak'ah has been missed, and he must perform one rak'ah in its place after the Imam's salutation. If he performs it, then his prayer is valid. If he does not perform it, then his prayer is invalid, and he must repeat it.