Disclaimer: I am licensed to practice law in the state of California. Therefore if your case in not in California, the information contained herein may not apply. This answer is provided as a public service and as a general response to a general question, it is not meant, and should not be relied upon as specific legal advice, nor does it create an attorney-client relationship.
Your question is a little confusing as to how many children you have, but I am presuming you had one child and a second was born after the divorce was filed.
When parents divorce, it is not unusual for the parties to agree to splitting or sharing the tax deductions. If one parent is not employed or does not earn very much, often the deduction will do more good if it is shifted. If a qualifying child is claimed as a dependent by both parents and the parents do not file a joint return, the child is normally treated as the qualifying child of the custodial parent, (the parent with whom the child resided for the longest period of time during the taxable year.)
Without a court order, the controlling source is Federal Internal Revenue Code (IRC). The (IRC) allows the dependency exemption and child tax credit to be shifted to the noncustodial parent when a child receives over one-half of the child's support during the calendar year from the child's parents who are either: a) divorced, b) legally separated, c) separated under a written separation agreement, or d) living apart at all times during the last six months of the calendar year. The child must be in the custody of one or both of the child's parents for more than one-half of the calendar year.
If you have primary custody and your ex pays less that ½ of the total expenses incurred by the children, your ex cannot legally claim the children unless you release the exemptions to your ex.
The Federal rules state that release of the exemption and child tax credit for any calendar year requires the custodial parent to sign "a written declaration (in such manner and form as the Secretary may by regulations prescribe)" that he or she will not claim the child as a dependent for that year. Also, the noncustodial parent must attach the signed declaration to the noncustodial parent's return for the applicable year.
In other words, in order for him to legally claim either child, or both, he must attach a written release that you signed the release. Typically, an IRS form 8332 (which you can get on line from the IRS) is signed and attached to the non custodial parent’s tax return. Alternatively "If the divorce decree or separation agreement went into effect after 1984, the noncustodial parent can attach certain pages from the decree or agreement instead of Form 8332. To be able to do this, the decree or agreement must state all three of the following.
1. The noncustodial parent can claim the child as a dependent without regard to any condition (such as payment of support).
2. The other parent will not claim the child as a dependent.
3. The years for which the claim is released.
The noncustodial parent must attach all of the following pages from the decree or agreement.
Cover page (include the other parent's SSN on that page). The pages that include all of the information identified in (1) through (3) above and, the signature page with the other parent's signature and date of the Agreement."
As to your question on custody, yes, he can file, but it would be unlikely that he would succeed. I suggest you contact a local Family Law attorney before your ex files so you can prepare yourself.