I did this research back in 2015 but I didn't realise I hadn't posted it here on Casebook (it's on JTR Forums) until someone just asked me about it:
On the third February 1877 at the Union Infirmary, Greenwich, Catherine Eddowes, a hawker, gave birth to a son, Frederick William and registered him on 21st Feb 1877. Unlike the births of Catherine and Alfred who were registered with the surname Conway and their father's name given as Thomas Conway, Frederick's father is unnamed.
Catherine entered the Greenwich Infirmary from the Woolwich Rd workhouse and returned there after the birth on 20th Feb.
To corroborate that this is Catherine Eddowes; Eddowes older children, Catherine and Thomas were also admitted to the workhouse on 23rd Feb 1877 and a note says their mother is in the workhouse.
The family appear in workhouse records many times (I've found over 20), using both the surnames, Conway and Eddowes. Catherine used Kate and Catherine in different combinations also. She seems to have kept different surnames for different unions and is always Eddowes in Greenwich, which may explain why Frederick was registered as Eddowes, rather than Conway.
Thomas Conway jnr is listed with the middle name Lawrence in one workhouse entry but like Neal shelden, I haven't been able to trace a birth registration for Thomas.
I have been unable to trace Frederick after 1878 nor can I find a death for him.
Eddowes seems to have had links to Mill Lane near the Woolwich Barracks. It was occupied by soldiers and their families in married quarters and also in 1871 there are several vans parked up with families of 'licensed hawkers' in the area. One entry says Eddowes 'Pesters on Mill Lane', another tells us her two children were abandoned there by their parents, while she was still with Conway.
I have since found that the older children, Catherine and Thomas, were sent to South Metropolitan schools on a couple of occasions when their mother was in the infirmary or apparently, thanks to Gary Barnett's find, while their mother was in prison. This was the equivalent of being taken in to care and coming in the 1870's suggests that Thomas Conway was not taking sole care of his children at this time.
Is anyone able to find out what happened to Frederick William? I drew a blank He may have been adopted and so untraceable. I haven't been able to find a death for him. One of Catherine's distant relatives also had a child named Frederick born in 1877 as far as I remember.