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If the father requests he will likely be granted joint custody rights so long as he can stay in New Mexico. The fact that he is not a legal citizen does not matter for his custody rights so long as the court does not perceive a threat that he leave the country with the baby without court approval.
It is really up to the father to pursue his legal rights. Is he on the birth certificate?
If he is not on the birth certificate he will have to pursue paternity rights before he could get some type of custody. As for child support, she would still be entitled to support even if he does not have a social security card. I agree that it may be hard to collect if he is working under the table but she can do what she can none the less.
With that said child support and custody do not go hand in hand. He can get custody even if he is not paying support. The courts view them as two separate issues.
He can get custody even if he is not a legal citizen. I cannot comment on the politics of it. However, the courts have firmly sided that the rights as parents have superseded citizenship issues. He will never be granted authority to take her out of the country against your daughters objection.
Sorry to give you such news. I think your strategy to make him actively seek parental rights is the best strategy here.
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