Is the action of the Imam, in interrupting the prayer to strike children, then returning to complete it, correct from a Sharia perspective, taking into consideration that one of the congregants completed the prayer on his behalf?

What the Imam did by interrupting the prayer and hitting the children is a grave mistake. Interrupting the prayer contradicts its sanctity. The Imam sinned by interrupting his prayer if it was an obligatory prayer. If it was a supererogatory prayer, he is sinful according to the Hanafis and Malikis, and it is disliked according to the Shafi'is and Hanbalis. As for his hitting the children, it is not permissible unless he is their guardian or authorized to hit them. He is liable if the child dies from his beating if the beating was without the guardian's permission. As for what the congregants did by appointing a successor after the Imam's prayer became invalid, it is a correct action as long as they did so before performing a pillar of the prayer without an Imam. This is the preponderant opinion. The Imam, after interrupting the prayer, restarts the prayer anew as one of the individuals in the congregation, and it is not permissible for him to build upon what he had done.