New Zealand tax residents are generally taxed on their world-wide income you derive. This includes foreign income even if it is simply funding an offshore credit card or is deposited in an offshore bank account. The types of funds that are deposited into these bank accounts vary and include income paid by a non-resident employer, overseas life insurance policies, superannuation schemes, or equity investments held in portfolio accounts. The fact that foreign withholding tax may have been deducted on foreign income does not mean that this income is no longer taxable in New Zealand.
The income is calculated under the New Zealand rules and must be returned as offshore income. In most circumstances, this calculation is conducted on a cash basis. However, if it exceeds certain thresholds as set out in the Income Tax Act 2007, the income might have to be accounted for under financial arrangement rules on an accrual basis.
If interest is being paid to an offshore lender withholding tax might be payable in New Zealand. This does not apply if the offshore lender has a branch in New Zealand.
Be aware that IRD are looking at undeclared offshore income.
As these are complex areas, you may wish to seek professional advice if you have offshore bank accounts.