David Cameron and his most senior civil servant, Sir Jeremy Heywood, have been criticised by an all-party committee of MPs over the way they handled the Andrew Mitchell "plebgate" controversy.

The public administration committee said Heywood, the cabinet secretary, should have challenged the claim in a leaked police log that Mitchell called officers at the gates of No 10 "plebs" after Cameron asked Heywood to investigate what happened.

But, in a report published on Monday, the committee also said Cameron himself should have ordered a much more thorough investigation. Heywood was "not the appropriate person to investigate allegations of ministerial misconduct", they said, and instead Cameron should have involved Sir Alex Allan, the independent adviser on ministers' interests.

It also criticised him for not accepting a recommendation from the committee, backed by a vote in the Commons, saying that the prime minister's independent adviser on ministers' interests should have the power to launch his own inquiries into cases of this kind.

Mitchell was chief whip when he had an altercation with police officers at the gates of Downing Street in September last year. He denied calling the officers "plebs" but eventually resigned after it became clear he had lost the confidence of Conservative MPs.

Heywood became involved after John Randall, the deputy chief whip, received an email from a constituent who claimed he was a member of the public and had witnessed Mitchell calling the officers "plebs". The email appeared to corroborate an account of the incident that appeared in the Sun newspaper and Heywood was asked by Cameron to investigate its reliability.

When he gave evidence to the committee, Heywood said his remit had been "very limited" and that he was not supposed to be conducting a full investigation into what Mitchell had said to the police. He also said that, having established there was doubt over the reliability of the email, he helped Cameron to decide Mitchell should not be sacked.

But the committee said Heywood should have done more to establish what had happened. "Regardless of what the prime minister had or had not asked him to do, on establishing that there were unanswered questions about the incident, Sir Jeremy should have advised the prime minister that these questions required further investigation and therefore a wider inquiry," the MPs said.

On the night of Mitchell incident, the Downing Street head of security and Cameron's principal private secretary spoke to the officer involved. A minute was taken, and the MPs said it was "surprising" that Heywood could not tell them whether it included the word "plebs".

It was "equally surprising" that Heywood did not try to establish whether the account of the police "log" published in the Daily Telegraph was accurate, the MPs said. "Sir Jeremy could and should have advised the prime minister to refer the allegations of ministerial misconduct to the prime minister's adviser for a fuller investigation. That he did not do so is regrettable.

"He could also have advised the prime minister that it would be appropriate to refer any doubt about the police account of the incident to the relevant police authorities to investigate in order to resolve any discrepancies and inconsistencies."

In a report published in March 2012, the committee said the independent adviser should have the power to initiate his own investigations without having to wait for a request from the prime minister. The Commons backed this recommendation in a motion passed in July without opposition.

But Cameron has not agreed to give the adviser this power, and No 10 has not officially responded to the recommendation, even though Whitehall departments are supposed to reply to select committee reports within two months. The MPs said this delay was "unacceptable".