I have to ask "how do you know Ada was illigitimate?"
Firstly any birth without a father's name listed, or where the father was not married to the mother, was regarded as illegitimate.
Secondly it states illegitimate twice on Ada's birth registration.
Maybe as Debra suggests - MM (Margaret Mary) was not the marrying type.
I don't believe for one minute that it is that simple. You have 4 children who were born legitimately to Michael Moore and Margaret according to their birth registrations. Then after this relationship broke down you have 2 children who were born illegitimately to Sydney Woodland. For all the births that were registered (4 legitimate, and Ada) a date and place for the marriage to Michael Moore was given.
There also has to be a reason why Margaret did not marry Sydney Woodland (though she took his surname and pretended that she was married to him). The impediment to a marriage to Sydney was still being married to Michael Moore.
There was no reason for Margaret not to have been married to Michael, both Margaret and Michael independantly state the same date and place for this marriage to have occured, this marriage stopped Margaret from marrying Sydney, and there was no reason for Margaret to mention a marriage to Michael when she registered Ada's birth, if anything there was every reason not to mention it.
Just as every birth was regisered (and in this case the example is the birth of Syney Jnr), and just as every death was not registered (and there are many many of these - just compare the cemetery index to the death index), not every marriage was registered. Not many marriages would have missed being regisatered by 1879, but it still did happen. All it took was for the minister to forget to sent in the paperwork.
Thinking aloud. It is possible that Michael & Margaret had a home wedding. The paperwork for a home wedding would be more likely to be forgotten to be sent in by minister than one performed in his church. (They did forget to sometimes send in paperwork for church weddings too though.)
