How can one atone for the sin of adultery with a woman who is not one's wife and having a child with her, knowing that the laws of the country prohibit polygyny and reject divorcing the current wife?

You must improve your repentance and be sincere in it, by feeling remorse, desisting from the sin, and resolving not to return to it. Allah will relieve your distress and forgive your sin. We advise you not to leave your wife and daughter if you cannot combine and have multiple wives. Preventing the harm of oppressing them takes precedence over gaining the benefit of appeasing your cousin. It is not permissible for any repentant person to marry his cousin unless she has repented from her sin. A child born of adultery is not attributed to the man who committed adultery with his mother, and you are not obligated to financially support him, for the Prophet, peace and blessings be upon him, said: "The child is for the bedding, and for the adulterer is the stone." If you wish to show kindness to this child and his mother, that is a righteous deed for which you will be rewarded.