Trent Boult has again proved the major threat to England, nabbing two wickets to leave England in all sorts of trouble at 132/3 at stumps on day four.
He got the key wicket of Joe Root, who had just got his half-century, at the day's close.
Neil Wagner added to his early wicket of Alastair Cook, drawing fellow opener Mark Stoneman into a clipped shot straight into the path of Boult behind square leg.
Boult made the breakthrough earlier in the day, with Cook nicking into the path of keeper BJ Watling to derail England's second innings at Eden Park. The fast bowler removed Alastair Cook in just the third over.
Earlier, New Zealand sent the visitors into bat after declaring on 427/8 with a lead of 369 runs.
Centurion Henry Nicholls finished unbeaten on 145 runs, his highest score in test cricket.
He brought up the milestone shortly before tea, reaching the break with Tim Southee at the crease.
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The fast-bowler made a workmanlike 25 runs before England skipper Joe Root removed him with a catch off his own bowling.
New Zealand had started the day on 233-4 with Nicholls and BJ Watling at the crease.
Watling could only add 13 runs to his overnight score before being caught behind off the bowling of Stuart Broad.
Colin de Grandhomme made a quick-fire 29, which included six boundaries before Craig Overton had him edge one to wicket-keeper Jonny Bairstow.
Day two and three of the test were largely marred by rain, with only 17 balls bowled yesterday.
New Zealand: Tom Latham, Jeet Raval, Kane Williamson (c), Ross Taylor, Henry Nicholls, BJ Watling, Colin de Grandhomme, Todd Astle, Tim Southee, Neil Wagner, Trent Boult.
England: Alastair Cook, Mark Stoneman, Joe Root (c), Dawid Malan, Ben Stokes, Jonny Bairstow, Moeen Ali, Chris Woakes, Craig Overton, Stuart Broad, James Anderson.