If you are the custodial parent (you have custody more than half the nights of the year) then you are the only parent with the automatic right to claim a child as a dependent. The other parent can only legally claim the child as a dependent if you give them a signed statement, and even then, you are still the only parent allowed to use the child to qualify for head of household status, EIC, or the dependent care credit.
As of today, you could file amended returns for 2019, 2020 and 2021. The amended return for 2019 would be due by May 17, 2023. If you miss that date, it would be too late to amend the 2019 return although you could still amend 2020 and 2021.
What I think you are saying is that you share a child with a non-custodial parent, and you allowed the other parent to claim the child as a dependent in certain years, but you never gave the other parent anything in writing.
If that is the case, then yes, you could file amended returns to include the child as your dependent, and you may be entitled to certain tax credits depending on your income, filing status, and other tax facts. However, when the IRS processes the amended return and sees that two parents are claiming the child as a dependent, they will send letters to both parents asking for details about your custody situation and proof of where the child actually lived. The parent with the better proof will be allowed to claim the dependent and the other parent will be required to pay back any credits related to claiming the child, with interest, and possibly a 25% or larger penalty.
So you first need to ask yourself, could you prove the child lived in your home more than half the nights of the years in question.
Then, you should take note of the fact that if there is a family court involved in your case, and if the family court has ordered that the non-custodial parent can claim the child in certain years, the other parent could go to court and file a dispute over the fact that you amended your tax returns and caused them to suddenly owe thousands of dollars of back taxes, interest, and penalties. If the family court finds that you violated their orders, they can punish you severely if they want to. I would never recommend violating a court order unless your attorney tells you it's ok.
