Single parent moving

I suppose it comes down to whether or not you can afford the childcare and if you can do sufficient hours in the job and afford a life for you all here ?

Your 6 year old would be in school generally 9am-3pm and depending on where you are you could enrol them for breakfast club and after school club so you could drop that child off maybe 7.30/8am and then pick up maybe 5/5.30pm. That would probably cost you maybe $20-$30 a day?

Your 3 year old would have to attend private kindy or go to a carer/childminder if you wanted to work full time.
Depending on the visa status your 3 year old could get a grant of 20 - 30 hrs per week towards kindy assuming you found a qualifying one. This does reduce the costs quite significantly but you will be looking at anything from $100 to $200 per week full time child care for 5 days and that all depends on which part of the country you are in. We were in Wellington and paid nearly $180 per week for 3 days full time but that halved when we moved up to Tauranga for the same hours.
A carer/childminder would be a lot less $'s but generally they would be looked after in the childminders home with up to 4 other kids without all the facilities of a kindy.