Your Kids Account is theirs, so any tax owed is theirs too. If your child receives more than $200 NZD in a year from income that hasn’t already been taxed (i.e tax is deducted from your interest on savings, so NOT included in the $200 threshold), they will need to file an IR3 form - or you’ll need to on their behalf. Any dividends they receive through Hatch will be counted to that total.
If they hit the threshold of $50,000 NZD invested overseas, they’ll also fall under the FIF rules (get more info here). They’ll also be a very well set up child, so a little tax calculation shouldn’t put you off!