What happens if a foreign woman visits the US with a tourist or work visa and gets pregnant by an American man?

If the American man wants to claim his child and fully support and continue a relationship with the foreign mother but can’t legally be married to her because he is already legally married, but his wife is fully supportive, can the foreign mother be given a permanent visa? The child would have claim as an American citizen and the father has rights, but the child also has rights to have the mother and obviously needs the mother. And if the man loves the mother and his wife is ok with her and fully supportive, it would seem that the government couldn’t legally separate the child from either parent. It seems cruel! What are the laws and how would this work out best? What’s the best solution?

What happens if a foreign woman visits the US with a tourist or work visa and gets pregnant by an American man?
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