Praise be to Allah.
It is well known in Islam that zina (unlawful sexual relationship) is haraam and is a major sin. The one who has fallen into this sin has to repent to Allah, may He be exalted, before it is too late. The prohibition of this evil deed is more emphatic in the case of a married man, hence the punishment for it is stoning to death.
See the answer to question no. 97884
If it is proven to you that your husband is committing adultery, either because he has admitted it to you or there is shar‘i evidence to that effect, then advise him to fear Allah, immediately end his sinful relationship with that woman, do a lot of righteous deeds, and seek good company who will encourage him to do good and obey Allah and to avoid evil and sin.
It is not permissible for a woman to marry a man who is known to have committed zina unless he repents sincerely. If a woman marries a man who has been committing zina, then she is sinning and her marriage contract is invalid. If a woman marries a man who is chaste, then after marriage he falls into zina, the marriage contract is not annulled by his falling into zina, but this does not mean that the wife should accept him as a husband if he does not give up this immoral action. Hence we think that in your case, if he does not give it up, you should not remain in the marital relationship with him. Rather you should hasten to end your marriage to him, by means of either talaaq or khula‘.
You should understand that if your husband continues to commit zina, it will have a negative impact on his family members, his wife and children, in terms of both their upbringing and their health. So do not think of staying with him if he does not stop and give up this sin.
As you are living in a non-Muslim country and the law in this country forces the adulterer to acknowledge the child and also to spend on the child and take care of him, in this case Islam does not oblige you to look after the child, take care of him or breastfeed him, even if he was your husband’s child from a permissible marriage, unless you do that voluntarily and out of kindness.
Ibn Qudaamah (may Allah have mercy on him) said:
The husband does not have the right to force her (his wife) to breastfeed his child from another woman, or to look after him.
It was narrated from Jaabir ibn ‘Abdillah (may Allah be pleased with him) that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Have you got married?” I (Jaabir) said: Yes. He said: “To a virgin or a previously-married woman?” I said: To a previously-married woman. He said: “Why not a young girl, and you could play with her and she with you?” I said: I have sisters, and I wanted to marry a woman who would be able to bring them together, comb their hair and take care of them.
An-Nawawi (may Allah have mercy on him) said:
This indicates that it is permissible for a woman to look after her husband and his children and dependents if she agrees to do so, but if she does not agree to do so, then no (she should not be forced to do it).
Sharh Muslim, 5/203
Wali ad-Deen al-‘Iraaqi (may Allah have mercy on him) said:
This indicates that it is permissible for a woman to look after her husband and his children, sisters and dependents, and that there is nothing wrong with a man asking his wife to do that, but it is not obligatory for her to do so; rather she may do that if she agrees to.
Tarh at-Tathreeb, 7/112
To sum up our advice to you:
If your husband has not repented and given up zina, then you should separate from him and leave him and his child. But if he has repented from that and you think that he regrets what he has done, and you think it most likely that he has mended his ways, then there is nothing wrong with you staying with him and we advise you to help him to look after this child and take care of him, in the hope that Allah may reward you for that and compensate you with good, and perhaps this child may become righteous instead of being left under the care of disbelievers and being raised by them.
And Allah knows best.