1. Either one of you can claim your child together (child #2), but not both. The one who does not, should not even enter the child in turbotax (the interview caused some confusion on this last year and I don't know if it was fixed.) In turbotax, make sure to select that the child lived with you "all year" (all year since alive, which is the correct answer).
2. If your bf pays more than half the total household expenses, he can qualify for HOH if he claims child #2 as a dependent.
3. Your bf can't claim your other child (child #1) as a dependent if you have more than $4050 of taxable income. In that situation, you are a "taxpayer" for 2016, and he being a non-related person can't claim child #1 if a parent has partial custody and is a taxpayer (even if you are willing to let him, he can't). Even if your income is less than $4050, your bf can't claim child #1 unless child #1 lived in the same home as your bf for all 366 days of 2016, AND you don't file any tax return, even to claim a credit or benefit. And even then child #1 does not qualify your bf to file as head of household.
4. The person who qualifies to file as head of household is the person who pays more than half the living expense of the household. That is usually but not always the person with higher income. However, let's say that most of his money goes to pay child support for a previous relationship, or pays student loans in his name. Then you might be paying more money toward household expenses even though you earn less.
So overall, your choices seem to be
a. BF claims child #2 and you claim child #1. Whoever pays more than half the living expenses can qualify for HOH and the other files as single.
or b. you claim child #1 and #2 and bf claims no dependents. bf must file as single and you also must file as single unless you really do pay more than half the living expenses, then you can file as HOH.
Which is better financially can only be determined by making test returns both ways.
