As an adoption professional, one of the hardest tasks we have is evaluating inquiries to assess a "fit" between applicants and the agency. Aside from specific criteria (i.e. age, length of marriage, infertility diagnosis), there typically isn't any one item on the inquiry that would deter us from approving the inquiry, rather a combination of factors. Ultimately when approving inquiries, we have to give first preference to families who make ideal placement resources for the children we are placing.
It is no secret that we place a large number of full Hispanic children here at Abrazo. So families open to full Hispanic children are always needed and space at our orientations is always saved for those couples. Obviously, families also open to a variety of other backgrounds including African American and biracial children are always desirable as placement resources as well. (Please note: children of Anglo/Hispanic background are not bi-racial--both are of the Caucasian race. "Biracial" denotes descendents of two races, ie., Anglo/African American, Hispanic/African American, Asian/African American, etc.)
In addition, we do limit the space that's available for couples already parenting. We find that couples with a child/children already in the home wait longer than first time parents-to-be, although birthparents are more receptive to placing with families with adopted children, vs. biological. Are there exceptions to this? Absolutely; however, in our experience a large number of our birthparents do prefer childless couples. Would the fact that you have a child be the sole factor in determining whether we approve or deny your inquiry? No; it would also depend on other factors, such as level of openness, geographic location, willingness to consider older children, etc.
I do realize how difficult it must be to receive a letter stating we are unable to proceed to the application step, but I hope that no family has deterred their dreams of building/expanding their family solely because we were unable to assist them. There is no shortage of adoption agencies out there and every agency decides their own admittance criteria, so hopefully with enough research everyone can find "the right" agency for them.