Hola, Caleb. In this case there is the regionalism of using the pronoun “me” when we are talking about another person (el niño), because the boy is considered someone close to the person talking, could be their kid. Without using “me” the sentence would be: No moleste al niño. (Don´t bother the boy). But with “me” it might be understood as: don´t bother my boy.
If we use “con”, making the sentence: “no me moleste CON el niño”; it means something like: don´t bother me with the boy. As in don´t interrupt what I´m doing to give me the responsibility to see or care for the boy.
Espero haberte ayudado. Soy tutora aquí, siempre puedes enviarme un mensaje o reservar una clase conmigo si deseas continuar practicando. Espero te siga yendo muy bien en tu práctica de español.