Kate was my grandmother's sister and probably had her own reason for keeping the name of her children' father secret.
This is quite misconceived. Kate Carr wasn't keeping the father's name secret. The father wasn't named because that was the (British) law. Illegitimate children legally didn't have a father!
So putative father was not supposed to be named, even if known (which was not always the case).
As you seem to know about the family, can you say where they were located on the 1901 and 1911 census? Might help to find baptism records, which may record father - if Catholic. If Protestant they will almost certainly follow British law and not name the father.