You can use HSA and FSA funds for medical care for yourself, a spouse, or a dependent.
Since you are living unmarried with the other parent of the child, only one parent may claim the child as a dependent on their 2017 tax return. That is the parent who can also pay for medical expenses from an HSA or FSA or take the medical expense tax deduction or child care credit for expenses paid with after-tax dollars. Assuming you and your partner will live together with the child for the rest of the year, then either one of you can claim the child as a dependent, but not both -- a child may only be claimed as a dependent once. If you will claim the child as a dependent, you can use the pre-tax money in the FSA or HSA to pay for expenses. If you don't claim the child as a dependent, you can't use the tax-free money to pay expenses or if you do, the expenses won't be "qualified" and you will owe income tax on the use of the money.
(The rules that allowing "splitting" of certain dependent benefits only apply when the parents are living apart and sharing custody.)
Exactly how you should claim the child on your tax returns for the lowest overall tax/highest refund is complicated, and you may want to test different scenarios. If the parent who pays more than half the household expenses claims the dependent, then that parent can also use Head of Household filing status instead of Single, which results in lower taxes. But if one parent is lower income and within the range for EIC, that parent might claim the child even though they have to file as single, and the EIC might offset the other parent's higher taxes.
Either way, only the parent who claims the child as a dependent can use the tax breaks for medical and child care expenses.
