How can one deal with a husband's betrayal that has led to the wife's alienation and hatred, given the presence of a child who connects them and the difficulty of separation, and how can one overcome the deep sorrow resulting from this betrayal?

The husband must sincerely repent from corresponding with and contacting the foreign woman. If he regrets and his repentance is true, then you should forgive him and forget his slip, for sincere repentance erases mistakes. You should also look into the reason he sought someone other than you, and be a good wife to him, beautify yourself for him, and show him affection, so that he may find chastity with you.