Manchester City’s plight in the Premier League means the double treble is unrealistic but their quest to become the only club to win both domestic Cups in successive seasons continues. Port Vale were duly defeated, even if a side 75 places below City in the league ladder acquitted themselves rather better than some more illustrious visitors to the Etihad Stadium. Watford conceded eight goals here in the Premier League, Atalanta five in the Champions League. Vale only let in four while their talisman, Tom Pope, found the City net. A starting 11 assembled at a cost of nothing helped bank some £300,000 and reputations were enhanced. Oleksandr Zinchenko, Sergio Agüero, Taylor Harwood-Bellis and Phil Foden nonetheless ensured the favourites progressed.

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City, often criticised for giving young players too few chances, could savour the sight of two teenagers and lifelong fans on the scoresheet. “Of course it is important for Taylor and Phil,” said Pep Guardiola, though he was quick to point out they were selected on merit, rather than for sentimentality. “It is not because they are English players. We don’t give presents.”

It nevertheless amounted to a day to remember for Harwood-Bellis. The 17-year-old became the youngest Englishman to score for City since Micah Richards, even if he knew little about his first senior strike.

Vale will recall it more for the actions of a man twice his age and with more than a century of goals to his name. Pope’s mischievous tweet last summer that he would score 40 goals a season if he were up against John Stones every week formed the backdrop. Guardiola set up a duel by recalling the England defender. Pope, however, showed a veteran’s nous by isolating himself against the rookie Harwood-Bellis to meet David Amoo’s cross with a brilliant header, angled beyond Claudio Bravo.

The City goalkeeper conceded to the opposition’s first attempt, something that happens so frequently it almost constitutes his party trick. Vale’s official Twitter account, perhaps not realising the new year honours list has already been announced, called for Pope to be given a knighthood.

He almost cemented a status as Stones’ unlikely nemesis, blocking the defender’s casual pass for the substitute Tom Conlon to put a diving header just past the far post. He gave their travelling support of 8,000 much to enjoy. These clubs rubbed shoulders in the second tier in 1999 and had not met since then as their paths have diverged. Briefly, Pope brought parity.

His antics apart, however, City were predictably dominant. Vale retained their shape well and the goalkeeper, Scott Brown, excelled, with fine saves from Foden, Riyad Mahrez and Bernardo Silva, whose header he tipped on to a post.

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City had 22 shots and ought to have scored more. David Silva has been a byword for precision but was responsible for a terrible miss, failing to convert an open goal after a centre from Ilkay Gündogan. The German’s passing verged on the majestic as he sprayed the ball around with elegant incision, but it took a stroke of fortune to give City the lead.

Zinchenko let fly from 25 yards but his shot took a crucial deflection off Leon Legge as it flew past Brown, coming off the Vale captain’s leg. Minor milestones followed for one of Britain’s oldest clubs; for the first time, a Vale game was affected by VAR decisions. Two marginal decisions brought City’s second and third goals.

Agüero bundled the ball in after Gündogan’s delightful chipped pass and Foden’s volleyed centre, with the youngster ruled just onside. Harwood-Bellis had been deemed offside when Stones’ shot diverted in off his shin, but the decision was overruled after replays were consulted. “It was a little bit of a weird goal,” said Guardiola, concerned that celebrations were put on hold while technology was consulted. “Two VAR goals and a deflected shot,” rued the Vale manager John Askey.

The fourth was simpler and smoother, Foden slotting in Angeliño’s cross. Come the final whistle, City invited the League Two players into their dressing room. “They have had a bit of a laugh, especially about Popey and John Stones,” added Askey. Not that the Vale scorer managed to swap shirts with the target of his taunts. “I managed to get Zinchenko’s,” he said. “The rest of them pied me off.”