“Accounting for” has a couple of definitions:
In this context it is talking about value.
<ol> <li>My rent is $500/ month, but if I “Take into account/ account for” my utility bills, my rent is actually $650/ month.</li> <li>My rent is $650/ month, but if I “take into account/ account for” my roommate paying half of the bill, my rent is actually $325/ month.</li> </ol>So an amount changes when you “account” for something. Accounting is addition and subtraction! It is not “reporting, understanding, or recording”. It is adding or subtracting value.
“failing to account for it”: It is not “understanding”. It is a change in value. A family has $1200 every month. But every month they have to pay $600 dollars for their house. Do they actually have $1200? No. You only think they have $1200 because you have “failed to account for” housing. It isn’t that you don’t understand, but you have not added it to the equation.
“more children are defined as poor after accounting”: It is not “reporting”. If I say, “Children are poor if their parents “bring home” less than $1000/ month.”
Not accounting for housing: $1200/ month: Not poor.
Accounting for housing: $600/ month: Poor.
“income is available after having accounted”: The key part of this sentence is “accurate understanding” of the “available income”. I can say, “he has $1200 every month to spend!” This is not “available income”. Why? Because he HAS TO spend $600 on housing.