New Zealand were 155-6 at the close, ahead by just four runs and with only four wickets remaining. Daryl Mitchell was not out 31 with Nathan Smith on one.
England made 499 in their first innings, taking an imposing 151-run lead, courtesy of a dominant 171-run knock by Harry Brook.
Captain Ben Stokes and Ollie Pope provided ample support to Brook with their dominant knocks of 80 and 77 runs respectively, followed by spirited cameos of Gus Atkinson (48) and Carse (33).
It was then the turn of England seamers, who wreaked havoc against New Zealand batters. The hosts’ salvage attempt faltered at the start with Tom Latham caught by Brook for one to give Woakes his first wicket in the third over.
Devon Conway was on eight when his attempt to pull Carse to the boundary was miscued to Atkinson at mid-on and New Zealand were tottering at 23-2.
Kane Williamson and Rachin Ravindra set about stitching the innings together as they put on 41 for the third wicket before Ravindra went for 24.
The left-hander could not resist pulling a short ball from Carse but failed to get on top of the bounce and was caught by Jacob Bethell waiting at mid-wicket.
Woakes returned to remove Williamson and Tom Blundell in successive balls to end the day with 3-39.
Williamson, so often called upon to rescue his country, had been unruffled until on 61 — his second half-century of the Test — he missed a straight delivery and was trapped plumb in front of the stumps.
With his next ball, Woakes found the faintest edge of Blundell’s bat but he was denied a hat-trick by Glenn Phillips who pushed a short ball towards the covers.
Phillips progressed to 19 before he was given out lbw to a Carse delivery that nipped back into the batsman.
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New Zealand sought a review but replays showed the ball was skimming the bails and the decision was the umpires call and Carse had 3-22.
Earlier, England resumed the day at 319-5 and raced along at six an over before being all out after lunch. A masterful Brook, who benefitted from five of the eight catches spilled by New Zealand, led England’s rescue after they were left reeling at 71-4.
The only good news for New Zealand was that while they dropped two more catches on day three, they successfully held five.
Brook smiled and looked to the heavens when Phillips missed a chance at 147, having been dropped the previous day on 18, 41, 70, and 106.
Four overs after his last reprieve, he brought up his 150 in imperious fashion, charging down the wicket at Tim Southee with the second new ball and driving him to the cover boundary.
But when New Zealand found a way to make catches stick, Brook’s innings, which included 15 fours and three sixes, came to an end.
A good length ball from Matt Henry found a faint outside edge and wicketkeeper Blundell grasped the opportunity.
The applause as Brook left the ground was as much for New Zealand holding a catch as it was for a magnificent innings.
Woakes (one) was out on the second ball he faced with Latham scooping up a low-level chance at the second slip.
Smith ended Atkinson’s breezy innings with Phillips taking a smart catch at long leg.
Henry was the most successful New Zealand bowler against England with 4-84. Smith took 3-141.
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