The UK foreign secretary is flying to Saudi Arabia to request that the Saudi crown prince, Mohammed bin Salman, cooperates more fully with the Turkish investigation into the murder of the Washington Post columnist Jamal Khashoggi in the Saudi consulate in Istanbul more than a month ago.

Jeremy Hunt will meet both King Salman and Prince Mohammed, the de facto ruler of Saudi Arabia and the man widely accused of ordering Khashoggi’s murder.

Speaking before a diplomatically fraught trip that includes a visit to the United Arab Emirates to try to broker a ceasefire in Yemen, Hunt said: “It is clearly unacceptable that the full circumstances behind his murder still remain unclear. We encourage the Saudi authorities to cooperate fully with the Turkish investigation into his death, so that we deliver justice for his family and the watching world.”

He added: “The international community remain united in horror and outrage at the brutal murder of Jamal Khashoggi one month ago.”

Hunt’s arrival will be the first meeting between a senior western minister and the Saudi royal family since the full circumstances of Khashoggi’s killing became clear, including the enforced Saudi admission that its intelligence agents killed Khashoggi in a premeditated murder.

On Sunday, the US secretary of state, Mike Pompeo, told Prince Mohammed that the US would hold accountable all involved in the killing of Khashoggi, in a telephone call that also took in the conflict in Yemen. “The secretary emphasised that the United States will hold all of those involved in the killing of Jamal Khashoggi accountable, and that Saudi Arabia must do the same,” spokeswoman Heather Nauert said in a statement.

Saudi Arabia insists Prince Mohammed had no knowledge of the plot to kill Khashoggi, and has arrested 18 people involved in the operation, including two of the prince’s closest aides.

Although the UK is making no formal read-across between Khashoggi’s murder on 2 October and the war in Yemen, it had been hoping that Saudi embarrassment would help it bow to pressure over potential peace talks in Yemen.

Yet the opposite has happened, with Saudi Arabia stepping up efforts to seize the strategic port of Hodeidah, making it more difficult for Martin Griffiths, the UN special envoy, to start peace talks.

Nevertheless, Recep Tayyip Erdoğan has been piling pressure on Prince Mohammed. On Saturday, he claimed that an audio recording of Khashoggi’s killing had been passed to Saudi Arabia, the US, Germany, France and the UK. “They have listened to all the conversations in them. They know,” Erdoğan said.

Turkish journalists, normally well briefed, said Khashoggi’s last words on the recording were: “I’m suffocating … Take this bag off my head, I’m claustrophobic.”

Saudi officials said initially that Khashoggi had left the consulate, later saying he died in an unplanned “rogue operation”. The kingdom’s public prosecutor, Saud al-Mojeb, has since admitted the journalist was killed in a premeditated attack.

The UK was fully informed of the content of the recording by the Turkish foreign minister at a meeting in London three weeks ago, but Erdoğan’s remarks represent the first time the existence of audio footage has been confirmed.

British sources did not officially confirm the UK had been passed a copy of the recording, but it is understood members of UK intelligence services have had access to its contents.

On Monday, the French foreign minister, Jean-Yves Le Drian, said France was not in possession of any recordings as far as he was aware, contradicting Erdoğan’s remarks“. Asked in a France 2 interview whether he was accusing Erdoğan of lying, Le Drian replied: “It means that he has a political game to play in these circumstances.”

Turkey has not revealed whether the tape implicates Prince Mohammed directly in the killing. His chief domestic aide, Saad al-Qahtani, was sacked two weeks after the assassination, as was a deputy intelligence chief. The prince insists he played no role and his defenders have insisted the hitmen over-reached in a bid to please their masters.

Hunt has said the UK response to the murder will in part be determined by the level of Saudi cooperation with the inquiry, and the credibility of assurances that such a killing will never be repeated.

Emily Thornberry, the shadow foreign secretary, said: “It will be deeply concerning if Jeremy Hunt’s visit represents yet more empty talk, when what we urgently need is concrete action to hold Saudi Arabia to account for the murder of Jamal Khashoggi, and bring an immediate end to their assault on the port of Hodeidah, which is threatening the lives of millions of Yemeni civilians.”

In an article in the Washington Post, Erdoğan effectively accused Prince Mohammed of ordering Khashoggi’s killing, by exonerating King Salman of any responsibility and yet saying it had been ordered at the highest level of Saudi society. Turkish media has also reported a second group of Saudi agents came to Istanbul to cover up any signs of the murder including using acid to dissolve the journalist’s dismembered body.

Irfan Fidan, Turkey’s chief prosecutor, said last week that Khashoggi had been strangled to death shortly after entering the consulate as part of a pre-conceived plot. The body was then dismembered and destroyed, he said.

Erdoğan has repeatedly accused the Saudis of continuing to dissemble about the murder by refusing to bring back the 18 arrested Saudi agents to Turkey, declining to identify a purported local cooperator that allegedly buried the body and by denying Turkish investigators full access to the Saudi consul general’s house and garden.

In Saudi Arabia and the United Arab Emirates, Hunt will raise the need to agree the terms of a UN resolution binding all sides in the four-year Yemen civil war and focusing on a ceasefire and a political settlement to end the war. The UK is hoping to table a resolution as early as Friday but needs Saudi agreement.

Additional reporting by Pippa Crerar