1. You don't need an agreement with your ex to claim your child as a dependent. If your child lives in your home more than half the nights of the year, then only you are automatically entitled to claim the child as a dependent and use the child to qualify for HOH. Even if you released the dependent claim to your ex by giving them a waiver form 8332, only you can use the child to qualify for HOH. The IRS goes by where the child physically lives, not what any court order or agreement might say.
2. Your current household consists of yourself, your partner, child #1 (with your ex) and child #2 (with your current partner). Only one person can be head of that household. That is the person who pays more than half the household expenses. It only has to be $1 more than the other person, but if you each pay exactly half, then no one can claim HOH status.
If you pay more than half, then you can file as HOH using child #1 as the dependent who qualifies you to claim HOH. Your partner files single. Either you or your partner can claim child #2 as a dependent.
If your partner pays more than half, then your partner can file as HOH using child #2 as the qualifying dependent, and you would file single claiming child #1.
(Your partner can never claim child #1 as a dependent on their return, unless you earned less than $10,200, and don't file a tax return, and your partner were to pay more than half the child's expenses, and child #1 lived in your partner's home for the entire year.)
