We are truly saddened to hear about the struggles your family is facing, but we are encouraged that you have reached out for prayer. We will pray for your son, his girlfriend, and their children, and we will also provide some biblical counsel and correction.
Firstly, we want to address the fact that your son and his girlfriend are not married. According to the Bible, they are living in fornication and are in sin. This is not God's plan for relationships or child-rearing, and it is important for them to understand that. It is never good for children to be born outside of marriage. God's plan is for a man and a woman to be married and then start a family within the safety and security of a lifelong commitment.
"Do you not know that the unrighteous will not inherit the kingdom of God? Do not be deceived. Neither the sexually immoral, nor idolaters, nor adulterers, nor male prostitutes, nor homosexuals, nor thieves, nor covetous, nor drunkards, nor revilers, nor extortioners will inherit the kingdom of God." I Corinthians 6:9-10 WEB.
While the primary issue is the sexual immorality, you've also described your son as narcissistic and self-centered, which are traits God tells us to cast off.
"Let nothing be done through strife or conceit, but in lowliness of mind, each esteem others better than themselves." Philippians 2:3 WEB.
We are praying for your son to be delivered from all his sins including self-centeredness and narcissism. He needs to repent and seek forgiveness from God and from his girlfriend. He also needs to consider a Godly marriage and have a Godly perspective on his life.
We will also pray for his girlfriend, that God would give her strength, wisdom, and patience as she cares for their children.
"But if anyone does not provide for his own, and especially his own household, he has denied the faith, and is worse than an unbeliever." 1 Timothy 5:8 WEB.
Most importantly, we pray that both your son and his girlfriend would come to a saving knowledge of Jesus Christ. It is only through faith in Jesus that they can be saved from their sins and be reconciled to God.
"That if you will confess with your mouth that Jesus is Lord, and believe in your heart that God raised him from the dead, you will be saved." Romans 10:9 WEB.
Let's pray:
Heavenly Father, we come before You in the name of Jesus, lifting up this family to You. We ask that You would convict both the son and girlfriend of their sins as they repent and turn to You for forgiveness and healing. We pray for true salvation to all in the household. We pray for deliverance, healing, and guidance by the Holy Spirit for both of them. Help the son and girlfriend to accept one another, be kind to one another, and forgive one another, even as God, in Christ, has forgiven them. Help them to seek a Godly marriage and raise their children according to Your will.
We ask this in Jesus Christ's precious name we pray. Amen.
Prayer Focus: God, Thank You. Thank You for loving me. God, I ask You in Jesus’ name bless me with everything that I stand in need of and everything You want me to have. Bless me to know You in truth, fall in love with You with all my heart, mind, soul, body, and strength and never fall out of love with You. Bless me to have an ever growing closer stronger, more intimate relationship with You. Bless me with the love, desire, strength, and the spirit of obedience to always delight myself in You, seek first Your kingdom, and Your righteousness. Help and strengthen me God to always respect and obey You. Bless me to trust You with all my heart, acknowledge You in all my ways, and lean not to my own understanding. 
