All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
First of all, your husband is obligated to sincerely repent to Allaah, if he really had had an illicit relationship with that woman because it is not permissible for a Muslim man to have an affectionate relationship with a non-Mahram woman outside marriage.
However, if your husband had married this woman with the consent of her guardian and the presence of two witnesses, then she has Islamically become his wife; for more benefit on the conditions of a valid marriage, please refer to Fatwa 83629.
The fact that this woman is a Christian is not a problem if he married her after she had repented (from Zina) because it is permissible for a Muslim man to marry a chaste Christian woman; for more benefit, please refer to Fatwa 83922.
Also, it is permissible for a Muslim man to marry a second wife if he is able to be just between the two wives; for more benefit, please refer to Fatwa 84411.
Besides, if this woman gave birth to this daughter six months or more after he had married her, then this daughter should be traced back to him because the jurists stated that if a man has a wife and she gave birth to a child in a period that is possible for this child to be from him, then this child should be traced back to him. The period that a child could be considered from him is six months from the time the marriage took place; for more benefit, please refer to Fataawa 137526 and 142115.
However, if she gave birth to this daughter less than six months after their marriage, then this daughter is born out of an illicit relationship, in which case, she should not be traced back to him, but she should be traced back to her mother as we clarified in Fatwa 84788.
In any case, since your husband mentioned that he conducted a marriage contract with that woman, you should not investigate the matter. As far as you are concerned, you are still a wife of this man, so you should fulfill his rights and he should fulfill your rights. Among your rights is that he should be just between you and his second wife in regard to staying overnight with both of you and spending on both of you. It is not permissible for him to be unjust to you in regard to any of your rights and we advise you to be patient with him and advise him in case he is wrong.
As regards him spending the night outside, if this is the turn of his second wife, then he did what he is Islamically required to do as he is obligated to be just in staying overnight with both of you. But if he spends the night out of the marital home without a valid reason, then this is not among having good marital relationship with one’s wife as required by Islam. Moreover, if this is for forbidden matters, then he is sinful and you should frighten him with the Punishment of Allaah.
Allaah Knows best.