I think that it might well be Henry + Harriet Robinson's sons that you are looking for.
This is a bit convoluted, bear with me!
We know that Arthur and William were admitted to the Greenwich workhouse in 1895, and that their father was said to be dead, and that their mothers address was 33 Reaston Street.
Looking at that family again, there are two school admissions to Colls Road Infants School in Peckham.
Arthur Robinson, admitted 4.9.1893 (but transcribed as 1898 on ancestry, though there is no doubt, and they have admissions from 1893 on the pages either side!)
Arthur's date of birth is given as 7 Dec 1887, parent or guardian is Harriett (mother), of 60 King Arthur St. No leaving dates given in the register.
So with the mother being named as the parent, it looks to me that Henry had died. Although on the second Colls Road Infants School record -
Alfred Robinson, admitted 24.8.97, born 29.9.92. Father Henry, address 34 Loder Street (no leaving date). So somewhat similar details to the baptism at St Jude (but birthday 6 days out, living at 34 rather than 31 Loder).
The son William Robinson born 1890 is not indexed at the school, but I am wondering that maybe he was not going to start until he was around about 5, and never made it there. There is no likely death record in south London that I can see.
Arthur Robinson does not appear to be indexed later as being admitted to the Boys School, even though his younger brother was admitted to the Infants in 1897.
So it's possible that William and Arthur are the ones going to Sutton - Harriet applies for relief, she keeps the youngest son Alfred, but the price of help is that the two older boys have to go into the workhouse school.
But why Greenwich?
On Arthur's school record it says that last school attended was Monson Road. Now that school is in New Cross, and just round the corner from Reaston Street.
And looking for a possible death of dad around the time of the change of schools (and parish/union), there is this one
Sep 1893 Greenwich 1d 679
And now we get lucky. He died in the Greenwich Union Infirmary.
Creed register, 1891-1893
admitted 10 May
parish St Pauls
relative - Wife Harriet 22 Reaston St
discharge - Dead 21.7.93
The A+D register gives Henry's occupation as Labourer.
And less than two months after Henry's death Harriet has son Arthur admitted to Colls Rd Infants, back in Camberwell. Perhaps she returned to Reaston Street for a spell in 1895?
So I am hopeful this will work out.
- Greenwich Infirmary Creed.jpg (16.52 KiB) Viewed 2097 times