I haven't been following this thread very closely so my suggestion may be totally irrelevant. At the very least it's pure speculation of course.

When she went to register the birth the registrar would likely ask the questions and write the answers in the order needed for the certificate - baby's date of birth, name, gender, FATHER'S name. When he then discovered that they weren't married - and presumably the father hadn't accompanied her - the registrar said oh dear, we can't have that! and crossed him out. Maybe?