Should a husband take back his wife who cheated on him, after a year of separation, in order to protect the reputation of his two daughters, especially considering the difficulty of raising them alone and his frequent travels, or should he marry another, despite his fears of her mistreating his daughters? What are the Shar’i conditions for taking her back and her punishment, and would he be considered a “dayyuth” if he takes her back?

Your wife erred by engaging in illicit relationships. Her actions should have been addressed with admonition, warning, and strong reprimand, not by abandoning her for a full year. Abandoning one's wife is not permissible except for a legitimate Shar'i reason, such as insubordination (nushuz). You must take her back into your home if she genuinely repents. You are permitted to stipulate that she must not return to any illicit relationship, and you will not be considered a "dayyuth" (one who tolerates indecency in his household) by doing so. A dayyuth is one who condones evil within his family. You are allowed to discipline her, first through admonition, then by abandonment in the conjugal bed, and then by light, non-severe striking, if she does not respond.