Is there a dhikr or du'a that helps with steadfastness and patience in dealing with children's mischief?

There is no specific dhikr or supplication known for such a situation. It is recommended to recite what is easy from the Qur'an, especially Surat Al-Fatihah, Ayah Al-Kursy, and Al-Mu'awwidhat (Surahs Al-Falaq and An-Nas), and some prophetic supplications, such as the supplication for seeking refuge for Al-Hasan and Al-Husayn: "I seek refuge in the perfect words of Allah from every devil and every harmful animal and from every evil eye."