Boris Johnson has said the Covid-19 testing system currently “has huge problems” during a sometimes downbeat appearance before an influential committee of MPs, as new figures showed nearly 4,000 UK cases in the 24 hours up to Wednesday morning, the highest daily total since early May.

With the number of people admitted to hospital in England with coronavirus also reaching the highest level since the start of July, Johnson warned that the Covid death rate was likely to start rising in the coming weeks.

Appearing before the Commons liaison committee, Johnson conceded that there was currently not enough testing capacity, and “many people are deeply frustrated”. He additionally warned that the much-touted plan for a return to more normality through the mass use of new, near-instant tests was “a long way off”.

While Johnson has repeatedly warned of the potential difficulties from a surge in infections following the summer, his tone was notably different to the more optimistic stance he took at his most recent Downing Street press conference a week ago, and appeared to acknowledge mounting public concern.

Asked about delays faced by people seeking a test, or long trips to testing centres, Johnson reiterated his promise to increase daily capacity to 500,000 a day by the end of next month.

“We don’t have enough testing capacity now because, in an ideal world, I would like to test absolutely everybody that wants a test immediately,” he said.

Responding to questions from the Labour MP Meg Hillier, the prime minister rejected her characterisation of the government’s test-and-trace system as a failure, but conceded there had been difficulties.

“Although it has huge problems, and although many people are deeply frustrated, as several colleagues have pointed out, I think that they have done a quite remarkable job in expanding that operation from a standing start,” he said. “Yes, there’s a long way to go, and we will work night and day to ensure that we get there.”

Asked by another Labour MP, Catherine McKinnell, about the idea of reviewing the ceiling of six on the number of people who can meet at one time, Johnson warned that strict measures were needed given rising infection rates.

The latest daily figures showed that as of 9am on Wednesday there had been a further 3,991 lab-confirmed cases of coronavirus in the UK. Other data showed that 153 people with Covid-19 were admitted to hospital in England on Sunday, the highest daily figure since 1 July.

Johnson said he was wary about the idea of more than six people meeting, even if that included children.

“It is, alas, a fact of the disease that it is readily transmissible between children and adults, and what we are now seeing is, unfortunately, the progression of the disease from younger groups who, as everybody knows, are much less prone to its worse effects, up into the older groups,” he said.

The incidence of Covid in those aged over 80 was now 12 per 100,000, double that of a few days ago. The PM said: “It is growing. And although the number of cases is obviously far smaller than it was in the spring, we must expect those infections, proportionately, to lead to mortality. That is the reality.”

Johnson was also notably cautious about the so-called moonshot plan for up to 10m daily tests, with results in as little as 20 minutes, allowing people who test negative to gather in crowds, unveiled last week.

Saying he did not recognise the 10m figure, Johnson said: “We are are long way off, I’m afraid, or still some way off, having those instant, pregnancy-style, liberating tests that tell you whether you are infectious or not.”