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New Zealand, who lead the two-match series 1-0, had established a second innings lead of 110 runs in Hamilton on Tuesday with Williamson on 96 while Ross Taylor was on 84 in an unbeaten 183-run partnership.
Taylor became just the second New Zealand player to surpass 7,000 test runs when he pushed a Joe Root delivery into the leg side for a single to reach 83 just before the lunch break. Stephen Fleming scored 7,172 runs.
Denly's spilled chance of Williamson, however, was the key moment in the first session on Tuesday that could have given England a chance of bowling the hosts out with a small lead.
New Zealand captain Williamson had advanced to 62 when he was fooled by a slower delivery from Jofra Archer and through his shot early with the ball looping to Denly at short mid-wicket.
Such was the confidence that Denly would snaffle the chance and open up New Zealand's middle order at 154-3, Archer was halfway down the pitch celebrating while several fielders raised their arms in jubilation.
Their excitement, however, quickly turned to shocked disbelief as Denly dropped the ball, while Archer could only bury his face in his hands and burst into laughter.
Williamson was also dropped earlier by wicketkeeper Ollie Pope while on 39.
New Zealand had resumed their second innings on Tuesday on 96-2, just five runs behind England's 476 that had been anchored by a majestic 226 from captain Root to break a personal slump in form.
Root also shared in a 177-run partnership with Rory Burns (101) on Sunday and 193 with Ollie Pope (75).
New Zealand won the first match at Bay Oval in Mt. Maunganui by an innings and 65 runs.