The only reason I can think they were registered in their mothers name was because she wasnt married to Frank,I am not saying they were not his children ,but if and a big if is ,if he is married to Philomena Verichi and they were seperated, and he lived with Agnes as man and wife ,and then he dies in 1931,still married to Philomena ,who would support Agnes and her children,I do think you need to find where your Grandmother was as a child and try and access her records,it may give quite a lot of detail for her admission into the childrens home,
this may be completly wrong ,but I have turned the Name inside out over the last few days,Benedito Fasano .Fasciano was living with the Verichi family in 1901 ,in 1911 he is head of household in Southport an ice cream maker,name and its spelling changes so many times,its hard to track even with phonetic spelings ,Joseph Fasiano, Bens sons war record states he goes by the name of Daniel,get my drift ,a lot of changing and chopping around of first names to,