The Hodgsons were living in the community, the deaths might have been in the workhouse - the Chorlton registration district covers a reasonable sized area in the south of Manchester.
To summarise:-
1901, you have Walter and William in the Cottage homes, part of Chorlton Union, put into the workhouse system by widowed mother Harriet. There is also a widowed Harriet b. Ripon in the Chorlton workhouse.
There are only 2 Walter Layfield births on FreeBmd between 1891 and 1901, one was born in Yorkshire and is with his grandparents in 1901, leaving the birth registered in Chorlton dec qtr 1898.
There is a registration for a William Layfield march qtr 1892 Liverpool vol 8b pg 103 - the correct quarter for someone born late December 1891 as you have 6 weeks to register a birth.
1891 census has a Layfield couple living in Liverpool, both born Yorkshire with a 12 year old son Allan, registered as Allan Hodgson Layfield in West Derby (Liverpool), and there is a marriage of William Layfield and Harriet Preston C Hodgson in Liverpool.
1858 there is a birth registered in Ripon area of a Harriet Hodgson, a little older than claimed in 1901, but workhouse ages aren't always accurate.
1861 sees Harriet Preston Calvert Hodgson with large family on farm in Laverton, Yorkshire, which would be covered by Ripon registration district
1871 sees Harriet with family in reduced circumstances in Manchester. I have found also in 1871 3 of her brothers living in Liverpool, 2 of them with families called Calvert headed by a cowman/dairyman. And in their census entries, they give Ripon as their place of birth.
1891 Harriet's parents are living in Chorlton upon Medlock. Isabella dies in 1891, possibly Harriet moves to Chorlton to look after her father. The family would have to be living in the area served by the Chorlton Union for them to be in the workhouse system there.
The problem I do have is with the boys' father's death. There is this death reg
William Layfield aged 44 dec qtr 1891 Liverpool vol 8b pg 107
which would mean that Walter had a different father or it's the wrong death. He was a drayman/carter so could have died in another area and if he wasn't known there, registered as unknown.
There are possible births & deaths for a sister also in Chorlton
Mary Layfield birth - sept qtr 1894 vol 8 c pg 746
death aged 3 dec qtr 1897 vol 8c pg 518
Annie Layfield birth sept qtr 1897 vol 8c pg 742
death aged 3 dec qtr 1900 vol 8c pg 453
Janet Layfield birth sept qtr 1891 vol 8c pg 696
death aged 1 dec qtr 1895 vol 8c pg 541
Once you get more info from Manchester, I'd say the birth certs for William & Walter should be first on the shopping list.