Living as a single mum in NZ is tough, unless you are high income. Been there, got the t-shirt.
Admittedly this was 15-20 years ago, and things have changed a bit for the better but things to remember, there is no child benefit, tax from memory tax credits are a maximum of about $5000 (although it may have gone up in the past year or so since Labour was elected) per year and per child if you are sole parent depending on your income.
Other factors - GP costs about $30 for an appointment unless you can be subsidised by the Govt, you have to pay school 'donations' of about $80 (not really donations in reality if you don't pay, school doesn't get income so most parents feel compelled to). There are no such thing as school dinners, unless you live in a very deprived area (South Auckland pretty much) that's when school lunches are provided, otherwise its packed lunches. Heating (there is no central heating either), food and petrol is bloody expensive.
I am not trying to be negative but just pointing out the reality that there is not the same safety net that is in the UK. I left NZ in 2004 simply because I couldn't afford to raise DD there after myself and exp and I split.
However it is amazing place to bring up children, sometimes I have regrets about leaving but unfortunately you can't pay the bills by the landscape / lifestyle