England is chasing a winning target of 296 to sweep the series after dismissing New Zealand for 326 at tea on Sunday with Jack Leach taking 10 wickets in the third test at Headingley.
The England spinner claimed 5-66, including the final wicket of Trent Boult (4), to go with his 5-100 in the first innings.
Tom Blundell ran out of partners and was 88 not out at the end of New Zealand's second innings. He shared another impressive partnership with Daryl Mitchell (56), who was trapped leg before wicket in the afternoon by pacer Matthew Potts (3-66). The pair put on 113 for the sixth wicket.
The tourists resumed the fourth day on 168-5, stretching their lead to 223 at lunch.
Mitchell and Blundell have been constant irritants to England over the course of the series, sharing big stands of 195 and 236 at Lord’s and Trent Bridge, respectively, and the pair once again combined to shut down the home attack. Mitchell became the first touring player to score more than 500 runs in a test series of three matches or fewer in England.
England had started the day with an unexpected personnel change when wicketkeeper Ben Foakes tested positive for COVID-19. He had already been absent on day three due to a stiff back.
Sam Billings was summoned as an emergency replacement, earning a second test cap after his debut in the final match of the Ashes.
Billings, who will also be able to bat in the fourth innings, was England’s second substitute of the series after Matt Parkinson stepped in for a concussed Leach at Lord’s.
England is hoping to sweep the three-match series against the world test champion. With the new emphasis on entertaining and aggressive cricket — England successfully chased down 299 with some 22 overs to spare in the second test — Ben Stokes' side has plenty of time.
But New Zealand will also be optimistic after reducing England to 56-6 in its first innings.
