While I don't understand what you mean by "majority," I assume it means most of them and they are specified in the agreement assuming there was no trial. If the agreement thereby allows you to claim them, then your ex-spouse should not be able to, and if you both claim, the IRS will flag and determine who gets the deductions. Typically, the agreement states that the other spouse, who could normally claim the exemption, annually sign the appropriate tax form enabling you to take the exemption pursuant to the judgment. Consult the attorney who handled your divorce or another divorce attorney in your area.
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If the children reside with you more than not, then you should be able to get the HoH deduction. If not, then only the dependant deductions are transferable per IRS rules. It is not clear from your post why the tax preparer denied you, but you may want to work with your ex- and find another accountant.
If your judgment states you get to claim the kids then you get to claim them. Be sure there is no other language that is limiting like stating if you are behind in child support then you cannot. I would have a lawyer review your judgment. The tax preparer, who obviously is not an accountant, does not know the law and is most likely advising you wrong. Get an accountant. Even if she wrongfully claimed them you can still legally claim them. Better yet go to a lawyer for review wo is an accountant also, Plenty of those. Tax preparers need not even be college graduates.
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You can ONLY claim Head of Household if the children lived with you for more days than your ex during the year. You can claim the exemptions and Child Tax Credits for the children if your ex had them a greater number of days, but your ex will need to provide a signed Form 8332.
If they children lived with you for more days, you get to claim them and there is no question. Go to another preparer.