What should I do about my suspicions regarding my husband's relationship with another woman, and how do I deal with my fear for my daughters concerning Allah's punishment?

The fundamental principle is to think well of a Muslim, especially between spouses. If the matter is merely suspicion, it should be abandoned. However, if an illicit relationship is confirmed, you must try to reform him to preserve the family. If he responds, then praise be to Allah. But if he persists in immorality and you are certain of it, then you have the right to seek a divorce. The husband must also be warned about the danger of his bad example to his daughters and keeping them away from places of temptation.