If I put that I had a child in 2020 but didn't claim him, why can't his mother claim him, I never claimed him as a dependant

If you are not claiming the child why is he entered on your tax return?  What were you trying to do?   Do you and the other parent live together or apart?    If all of you live together then the child should only be entered on one of your returns. 

 If you live apart then you can sign an agreement   (Form 8332)  that allows the custodial parent (the parent the child spent the most nights with) to file as head of household, get childcare credit and earned income credit.   The non-custodial parent gets the child tax credit.

If the two of you have had some mix up with the rules for claiming your child  (and the rules are confusing, for sure)   the other parent who is claiming the child will now have to print , sign and mail her return that claims the child.  She will not be able to e-file this year even if you amend.   You cannot amend to remove the child until your return has been fully processed and when the Form 1040X becomes available at the end of March.

If you put your child's information on your income tax return, you may have accidentally claimed him as a dependent.  

If you want to remove the child, you must now file an amended income tax return.  

In order for the mother to claim the child, she will have to file her income tax return by mail.