Up to 60 percent of beneficiaries at seven centres in Sanaa "confirmed they had not received any assistance" and 33 percent of respondents in the rebels' northern stronghold of Saada received no food in April, says head of the UN food agency.
As part of the "phase one" of a ceasefire, the Houthi movement's unilateral withdrawal from the key ports began on Saturday, in the most significant advance yet for efforts to end the four-year-old war and relieve hunger.
Mohammed Ali al Houthi, the head of the rebels' Supreme Revolutionary Committee, says the pullout started on Saturday at 0700 GMT. But the Yemen government says the rebels are faking withdrawal.
As the conflict entered its fifth year on Tuesday, figures showed at least half a million children have dropped out of school with many doing odd jobs to sustain themselves and their families.
Warring parties exchanged fire on Sunday into Monday as the UN seeks to prop up a ceasefire in the key port city Hudaida, a lifeline for millions at risk of starvation after five years of bombing of Houthi rebels by a Saudi-led coalition.
Norwegian Refugee Council says sniper and explosive attacks have been growing despite an easing of air attacks amid a three-month-old truce in Hudaida, as the devastating conflict intensifies elsewhere.
Fresh fighting comes days after Yemeni government accused Houthi rebels of breaching ceasefire and refusing to withdraw from port city in line with December agreement.
Significant donations came from Saudi Arabia and the United Arab Emirates, the two main countries in a coalition supporting the government side in Yemen's conflict.
Three months ago, UN Chief Antonio Guterres announced a series of breakthroughs, which are apparently yet to come into effect as a new UNICEF report says at least eight children are still killed or injured in Yemen every day.
Houthi rebels say their leader, Abdul Malek al Houthi, met with Griffiths on Sunday to discuss the implementation of peace deals from December talks with Yemen's internationally recognised government.
Commercial drones are increasingly being employed as a tool of political violence - will they be the new guerilla weapon of choice?
Retired Dutch General Patrick Cammaert chaired the meeting aboard a UN vessel docked off the coast of the flashpoint city of Hudaida.
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