I do agree with a huge amount of what you're saying, I do think there is some merit in his report though.
It is sad but true that many parents can't afford to feed their children healthily, and have problems paying for accommodation. BUT, I think that is a separate issue and should be treated as such really - I don't see money for helping with food/accommodation/clothing as being money to 'help social mobility' - it is survival money.
In terms of helping children move out of the poverty trap, I think money given to parents may often be misdirected - there are certainly a significant number of parents who cannot help with, say, reading, because they themselves cannot read. So I would much rather see the money going into a more targetted approach in the early years. I think a great many parents would take up an offer of 'free reading support' (for example) from age 3, who wouldn't necessarily spend money given out to them by the government on books/reading. (And of course there are many parents who WOULD - but I'm thinking about the children who are currently let down by lack of support, parental or otherwise, not the many children whose parents do a fantastic job irrespective of wealth).
So I think one lot of funding needs to come from the welfare budget - ie money for housing/food/clothing/heat etc. And the other lot needs to be under the Education department umbrella, and the two must be separate. Then you avoid the problem of providing reading to a family that now can't afford to buy food, which I agree would be a strange priority.