This has potential to become a very complex matter because of the consideration on international relocation.
Taking a child abroad without consent of the other parent is a criminal act of abduction and the child's father would be able to pursue return of the child to Australia under the Hague Convention, which is an international treaty protecting children and to which both Australia and New Zealand are signatories. I don't recommend taking that course of action. It becomes very unpleasant very quickly, and it is a violation of your child's right to know, spend time and communicate with both of her parents.
Instead, I would recommend separating from your partner and trying to keep things on amicable terms, because all future actions will be infinitely easier for both you and your child if things remain amicable. Maybe some counselling with Relationships Australia will help you on this path.
Once separated, you will be entitled to child support payments and may be eligible for additional payments from the government, and given that you will no longer be dependent on the finances of your de facto, his income will not weigh the means test down. One of the payment is the child care benefit, which will bring the cost of child care down significantly. Contact the Department of Human Services for information about this, or visit their website. I'm actually not sure what impact your status as NZ citizen will have on this, but given that you have a child who I assume is an Australian citizen, there may be some leeway.
I would also advise contacting Legal Aid and organising an appointment with a Legal Aid solicitor for legal advice. They offer consultations for free and will provide legal advice that is specific to your situation.
Shortly after separating, you might want to organise a family dispute resolution conference to sort out a property settlement and care arrangements for the child. Again, Relationships Australia and Legal Aid can help you with this.
I hope this helps. Remember that your daughter's needs come before yours, and one of those needs is maintaining a positive, healthy relationship with both you and her dad. What's best for your emotional needs may not be best for your daughter's, and there have been many ugly custody battles lost because of a parent's inability to separate the two. There are plenty of counselling and support services around as well that give you some guidance.
