We are truly concerned for your cousin's situation and we stand with you in prayer. First and foremost, the cure to really help them involves the marriage between the cousin and the child's father if that is an option since the child deserves to have both parents to raise them. Make sure to be very straightforward about the fact this is the best option for the child and the parents as well. It has been fairly obvious in society the detrimental effects of not having a father in the child's life. Most likely there is forgiveness involved. Remember Jesus said in the Sermon on the Mount
Matthew 6:14-15 WEB
For if you forgive men their trespasses, your heavenly Father will also forgive you. But if you don't forgive men their trespasses, neither will your Father forgive your trespasses.
Also remember
Matthew 18:21-22 WEB
Then Peter came and said to him, Lord, how often shall my brother sin against me, and I forgive him? Until seven times? Jesus said to him, I don't tell you until seven times, but, until seventy times seven.
We must also address the reason why you used the words 'child's father' rather than husband or spouse. Which means they are not married and their first mistake was having a child out of wedlock. This is unacceptable if either person involved knew better or were raised better. We have to address these such as they need to repent from this and strive to properly become a family.
Sounds like her daughter's father really needs to be disciplined by a godly man. Ask your cousin if you can possibly intervene to help fix the relationship. If this is not possible or the cousin refuses, then maybe she just needs to move on and not go back.
Matthew 18:15-17 WEB
Moreover if your brother shall trespass against you, go and tell him his fault between you and him alone: if he shall hear you, you have gained your brother. But if he will not hear you, then take with you one or two more, that in the mouth of two or three witnesses every word may be established. And if he shall neglect to hear them, tell it to the church: but if he neglect to hear the church, let him be to you as a heathen man and a publican.
Let us pray (Do not be alarmed that I have not mentioned the name of Jesus yet. I will address this please bear with me.)
Dear Heavenly Father,
We lift up the cousin and the daughter's father to you right now. We ask, Father, that You would give them the wisdom and strength to do what is right. Lord, we pray that You would guide their footsteps down the right path, and that they would turn away from following worldly desires, but to fully desire You in their lives. Help them to seek after You as if they were thirsty.We pray that you can unite them in marriage to be a family as you intended. Help them to realize they need to give up their selfish desires to look out for their child. We also pray that you open doors that no man can shut that gives them opportunities to make amends. In the name of your son.
Lastly, we need to address the reason we always pray in the name of Jesus Christ for any prayer. The scriptures say that there is no other name on which we can be saved.
Acts 4:12 Neither is there salvation in any other: for there is none other name under heaven given among men, whereby we must be saved.
Acts 4:18-19 And they called them, and commanded them not to speak at all nor teach in the name of Jesus. But Peter and John answered and said unto them, Whether it be right in the sight of God to hearken unto you more than unto God, judge ye.
Let us hear the words of Christ Jesus
John 14:14 If ye shall ask any thing in my name, I will do it.
Now that we see the words of Christ Jesus, why on Earth would be not as the name of Jesus Christ on our prayers. What better promise do we have that our prayers will be answered when we pray through the name of His Son.
Let us pray again,
Dear Lord, we thank you for your Word and we ask that you do do great things through us as your obedient servants that we may shine a beacon of light in a dark world to all unbelievers. Grant us hearts that we truly always want you first and foremost and that truly all we think about is bringing others to Christ. In Jesus name, we pray amen.