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Latham, on eight, was given his marching orders at the Bay Oval ground when rapped on the pads by Sam Curran’s third delivery.
But as he left the field, replays showed the ball scraped the inside edge of the bat, which would have earned a recall had he questioned the umpire’s decision.
At tea, New Zealand was one for 50 with Kane Williamson on 23 and Jeet Raval on 10 in reply to England’s 353.
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Latham’s dismissal was the second time in the Test that New Zealand has mistakenly failed to seek a review.
When England opener Rory Burns was on 10, they had an caught-behind appeal turned down although replays showed there had been an edge. Burns went on to make 52.
England, who resumed day two at 241 for four, added a further 112 as Tim Southee and Neil Wagner combined to take five of the last six wickets.
Southee, who claimed the key scalp of Ben Stokes for 91, took three wickets in 11 balls including Ollie Pope (29) and Curran with successive deliveries.
It reduced England to 286 for seven with Jofra Archer avoiding the hat-trick but only lasting three balls before he fell to Trent Boult.
Jos Buttler (43) and Jack Leach (18 not out) kept the scoreboard moving with a 52-run partnership for the ninth wicket before Wagner closed the innings with the wickets of Buttler and Stuart Broad in quick succession.
Southee finished with four for 88 and Wagner took three for 90.