We understand your concern for your friend and we will pray together for their situation. First, let's remember what the WEB says in Amos 3:3: "Can two walk together, unless they have agreed?" As believers, we should not be unequally yoked with unbelievers, as this can lead us away from our walk with Christ.
Your friend is in a relationship with someone who does not believe in God, and this may be influencing their choices. Let's pray that God will reveal Himself to both of them, and that your friend will return to their Christian roots. We should not be conformed to this world, but transformed in our mind to reflect Christ in Romans 12:2. Also, in 2 Corinthians 6:14, it says, "Don't be unequally yoked with unbelievers, for what fellowship have righteousness and iniquity? Or what fellowship has light with darkness?"
Regarding the issue of the gay cousin, remember that we should love everyone as Christ loved us, but we should not condone sin. We pray that this person also comes to know Christ fully and turns away from their sinful lifestyle in Romans 6:23.
We also see signs of rebellion in your friend's life, such as getting piercings without parental permission and wasting money on bad clothes and distant vacations. This is not honoring to their parents or to God. Let's pray that your friend will understand the value of honoring their parents as outlined in Ephesians 6:1: "Children, obey your parents in the Lord, for this is right."
We must also address the issue of sin in your friend's life. You mentioned that they have done some bad things. We pray that your friend will repent of their sins and turn back to God. Remember, in Acts 3:19, it says, "Repent therefore, and turn again, that your sins may be blotted out, so that there may come times of refreshing from the presence of the Lord." This is only by the name of Jesus Christ that we can repent and be saved as in Acts 4:12.
Let's pray together:
Dear Lord Jesus,
We come before You today to lift up our friend. We ask that You reveal Yourself to them and draw them back into a right relationship with You. We pray that You will open their eyes to the influences around them and give them the strength to walk away from anything that is not honoring to You.
We ask that You provide godly influences in their life, that they may find a spouse that loves You and will encourage them in their faith, avoiding adultery, fornication and all sexual immorality. We pray that You will give them the spirit of wisdom and guidance, and help them to live a life that honors You, their parents, and the principles outlined in Your Word.
We also pray for the cousin and the unbelieving significant other, that You will reveal Yourself to them and draw them to repentance and faith in You.
Thank you for all of the blessings that you give and for the love you have for us. Thank you for hearing our prayer and for the love you have for the friend. Help us to show the love Jesus commanded us to have in John 13:34.
In Jesus' name, we pray. Amen.
Prayer Focus: God, Thank You for loving me. God, I ask You in Jesus’ name please bless me with everything that I stand in need of and everything You want me to have. 
