Is it permissible for a worshiper during congregational prayer to extend his hand to stop a child running in front of the worshipers, citing a Hadith about the obligation to prevent anyone who tries to pass in front of him, and was his action right or wrong?

There are two issues here: The first relates to the worshiper (ma'mum) pushing away someone who passes in front of him. There are two opinions among the scholars regarding this, and some consider it permissible, stating that the Imam's sutrah (barrier) serves as a sutrah for those behind him in terms of preventing the prayer from being invalidated by the passing of a black dog. Therefore, there should be no denunciation in matters of ijtihad (independent reasoning). As for the second issue, the movement of pushing away a passerby is slight and does not affect the validity of the prayer.