A dear friend of mine who is a refugee married an American citizen from IL when she was 5 months pregnant (still living in Jordan at that time- they were married double-proxy.) The husband's name is on the son's birth record from Jordan, but after the baby was born, he abandonded the marriage.
My friend is now living in IL and is filing divorce papers.
Since the father of the baby lives in AZ, would it be better for her to pursue paternity through him? Her son has no US citizenship papers without a father listed. The only proof she has of her husband's consent to raise the baby is a copy of the birth record from Jordan with his name listed as the father. Is that enough to force him to pay child support and/ or sign citizenship papers for the baby?

If they were legally married at the time of birth, then he'd be legally presumed to be the father and no further evidence should be necessary. However, given the unusual circumstances of this particular case, if he attempts to contest paternity she may need a DNA test to prove that he is the father before he can be ordered to pay child support.