Yemen’s fragile ceasefire is holding and Saudi Arabia remains intent on reaching a negotiated end to the four-year-old civil war, Martin Griffiths, the UN special envoy for the country, has said.

Admitting the state of the ceasefire looked dire from the outside, he nevertheless said the key metric for the UN was the absence of offensive military operations to take territory and the end of Saudi airstrikes in the area.

Griffiths has been in Yemen’s capital, Sana’a, and the Red Sea port of Hodeidah this week to discuss blockages to agreements reached in UN-led talks in Stockholm in December. Yemen has been gripped by civil war between Iranian-backed Houthi rebels and the Saudi-backed – and UN-recognised – Yemen government of Abd Rabbu Mansour Hadi since 2015.

Griffiths said the vital next steps were gaining access to grain in Hodeidah’s mills, and a UN-sponsored meeting between the warring factions to start the process of redeploying Houthi troops.

Speaking on BBC Radio 4, he said the UN world food programme needed access to the mills in which enough grain to feed nearly 4 million Yemenis for a month had remained since October. Houthis claimed on Wednesday they were fired on by government forces as they tried to de-mine the route to the mills.

Griffiths also said he had plans for the UN-led redeployment co-ordination committee (RCC), bringing together the rival military leaderships, to restart its meetings within the next few days.

The Houthis recently refused to attend the RCCmeeting as it was due to be held in Yemen government-held territory.

Griffiths refused to disclose the proposed venue or agenda for the next critical meeting but said: “It is the redeployments out of the port and out of the city which are the essential aim of the Stockholm agreement – to demilitarise the entire port and city area. If we don’t, the ceasefire will inevitably fray and disappear.”

The Houthi forces are reluctant to withdraw from the city and port, and allow a new security force to take over. The nature of that security force is disputed and was not spelled out in the Stockholm agreement.

Griffiths said: “The Saudis are incredibly helpful in trying to make these things work.” He added: “At the political level we have the will – making this happen operationally on the ground – the first time ever these two sides have promised to disengage – is complicated and one bullet can change somebody’s life. It is tricky, it’s not perfect but we have to move forward.”

Some limited prisoner exchanges have started.