Possibly.
First, you have to claim a qualifying person. A qualifying person is a child dependent who lived in your home more than half the year. If you only separated after July 1, then each child lived in both your homes for more than half the year (since for much of the year, it was the same home). You can each claim a child as a dependent and they qualify you for HOH. However, if you have been living separately longer, then you need to determine whether the child(ren) slept in each parent's home at least 183 nights of the year (more than half the nights). If the children lived mostly with one parent and did not spend 183 in the home of the other parent, then that parent can't use them as qualifying children for HOH even if you have an agreement that allows you to share the dependent exemption. (The dependent exemption and child tax credit can be transferred by agreement but qualifying for HOH always stays with the parent who had physical custody more than half the nights of the year.)
Then, you have to pay more than half the cost of keeping up your home where the qualifying person lived. You consider "your home" over the entire year--did you pay more than half the cost. Suppose each parent is paying 100% of the cost of keeping up their post-separation home. You still need to look at who supported the home when you were living together, and the total costs of the household pre- and post-separation, and determine whether each parent paid more than half the overall total of their home costs for the year. So it will depend on when you separated and what your housing costs actually were.
