No and maybe. If the child is the father's dependent, no you cannot use those funds. HOWEVER, if the father is allowed to claim the child's exemption, that does not necessarily mean the child is his dependent. If you are the custodial parent, yes, you can use the funds for that child. Since this is a tax issue, the fact that the child's expenses would be allowed for the "Child and Dependent Care Credit" would be logical reasoning that the same would be true for Flexible Spending Accounts.
"Dependent care expenses cannot be paid to anyone
who is your child or stepchild under the age of 19 and claimed as a dependent
on your tax returns.
A dependent is defined as someone who spends at
least 8 hours a day in your home and is one of the following:
A tax dependent
child under the age of 13 for whom you have custody more than half of the year
. A dependent that is physically or mentally
incapable of self care regardless of age."
https://www.fsafeds.com/enroll
"Child of divorced or separated parents or parents living
apart.
Even if you can't claim your child as a dependent, he or she
is treated as your qualifying person if:
The child was under age 13 or wasn't physically or mentally
able to care for himself or herself,
The child received over half of his or her support during
the calendar year from one or both parents who are divorced or legally
separated under a decree of divorce or separate maintenance, are separated
under a written separation agreement, or lived apart at all times during the
last 6 months of the calendar year,
The child was in the custody of one or both parents for more
than half the year, and
You were the child's custodial parent.
The custodial parent is the parent with whom the child lived
for the greater number of nights in 2017. If the child was with each parent for
an equal number of nights, the custodial parent is the parent with the higher
adjusted gross income. For details and an exception for a parent who works at
night, see Publication 501.
The noncustodial parent can't treat the child as a
qualifying person even if that parent is entitled to claim the child as a
dependent under the special rules for a child of divorced or separated parents."
https://www.irs.gov/publications/p503#en_US_2017_publink1000203372
