So the couple were married 11 June 1906, and by 4 July 1906 the bride had

already involved the police in a search for him, obviously they did not find him locally, and so

she had proceeded, faced a Magistrate, explained her concerns, and

the Magistrate approved for a warrant issued for the arrest of her husband,
So by 4 July the Penrith police had already forwarded their paperwork to Sydney for printing in the NSW Police Gazette ... to that point in time, he was still being considered as a living person .... and likely still within NSW ... (Arrest desirable within the State)

The charge was to be 'wife desertion' and no mention of a charge of 'child desertion' nor any court order requiring him to pay maintenance ... perhaps insufficient time lapsed between 11 June 1906 and the date of the warrant (earlier than 4 July '06) for there to be such court order for maintenance or more likely ... a court order for maintenance may have needed a warrant to be issued to find him in the first instance...

Also, err ... perhaps the marriage was not ever consummated ...
A Question :
when did the suggestion he was in America first get raised ... JM