If you’re planning more than two stops in New Zealand in any 28 day period, or more than eight takeoffs/landings in a 365-day period, you’ll need to apply for a Foreign Air Operators Permit.
Otherwise, you can apply five days in advance for your single flight, or 30 days in advance for your series of flights. Apply directly to the authorities.
Auckland Oceanic FIR also requires no overfly permit, unless you’re overflying an island nation – check for their permit requirements.