This is a guarantee it won't fall under this amount. With sole parent support, family tax credits, and other support like accommodation supplement, you might be surprised how much some sole parents are getting now. With one child on sole parent benefit you'd be approaching $700, and that's without accommodation supplement.
Rather, kids are really expensive. You don't realise until you're living it but just things like how it's not practical for a single parent to flat with others, especially if the kids are young. Kids eat - often more than adults do (even young ones). And if you want a job you need to somehow pay for daycare, which can be $12k+ for one kid under 3 (I guess under 2 now, depending on when the promised change kicks in).
So let's say you live in Ōtara (I'm assuming a cheaper area of Auclland) as a single parent with one kid. Median rent for a 2 bedroom house is $575 so your money is almost gone already.
$150 a month for power (more if you have a cold house), $100 for internet/phone, and say $50/m for all insurances you need (contents + car). So that's say $80 a week, so we are at $655 and that's just the house.
Now we need food. This site says $200 a week for single parent with dependents. It doesn't break it down so we can assume 1 kid is probably cheaper but also it's from 2021 and food is a lot more expensive now so lets say it's about right.
Kids also need clothes and shoes they are constantly wearing out or growing out of, they need to be socialised (transport to park at a minimum), they need car seats (they grow, so need new ones), medical care is mostly free but again transport. Lets go low and call it a total of $20 a week.
We have now got $875 of basic expenses with no discretionary spending, on about $650-$700 of benefit.
Long story short, if we as a country want to help some of our most at risk kids grow up to be model citizens, $700 is less than the bare minimum.
In reality WINZ is constantly dealing with extra requests for help paying for things, especially larger one off expenses.