Is it permissible for a father to take his children and travel with them to an Arab country for fear of their being influenced by their irreligious mother who is teaching them an incorrect doctrine?

If the mother is unfit to custody her children due to her open transgression (fisq), then it is incumbent upon the children's paternal uncle to prevent them from being negatively influenced by her. It is impermissible to leave the one under custody in the care of someone who will not protect and nurture them. If there is no way to protect them except by traveling with them, then there is no harm in doing so for the sake of preserving their faith. He may allow them to communicate with her if there is no fear of temptation (fitna).