Saudi air strikes have been targeting Houthi rebels for almost two weeks.
The International Committee of the Red Cross (ICRC) was given permission by a Saudi-led coalition to land planes carrying staff and medical supplies.
The passenger plane landed safely on Monday but the supply flight has been unable to depart.
ICRC spokeswoman Sitara Jabeen says they had been unable to find a cargo plane to take their 48 tonnes of medical supplies into the country.
"Less and less airlines are either allowed, or able, or willing, to fly to Yemen," she said.
Ms Jabeen added that the country's security situation made finding a solution "extremely difficult" but that they were working to sort out the logistics.
Yemen: who is fighting whom?
The Houthis: Zaidi Shia-led rebels from the north, who seized control of Sanaa last year and have since been expanding their control
President Hadi: Fled to Saudi Arabia after rebel forces advanced on his stronghold in the southern city of Aden
Al-Qaeda in the Arabian Peninsula: Seen by the US as the most dangerous offshoot of al-Qaeda, AQAP opposes both the Houthis and President Hadi.
Islamic State: A Yemeni affiliate of IS has recently emerged, which seeks to eclipse AQAP
Failure 'not an option for Saudis'
Yemen crisis: An Iranian-Saudi battleground?
Meeting the Houthis - and their enemies
The rise of Yemen's Houthi rebels
The Red Cross is also trying to deploy a team of surgeons to the battle-torn city of Aden, but said that "authorisations from all the parties involved" were necessary before this could happen.
The ICRC spent a week negotiating with the Saudi-led coalition over deliveries of supplies.
It has called for a 24-hour ceasefire in Aden, while Russia has also urged the UN Security Council to support a "humanitarian pause" in the air strikes.
Explosions on Monday shook homes in the suburbs of Aden, one of President Hadi's last strongholds.
At least 53 people died in 24 hours of clashes between rebels and pro-government fighters, AFP reports.
Food, water and electricity shortages have also mounted, with residents pleading for help to feed their families.
Student Nisman Usman said: "We have lived three days of horrors - gunshots everywhere."
Arab views of the fight for Aden range from pro-Saudi optimism to fears of other players being dragged in.
Saudi and Gulf media report that the "shell-shocked" Houthis are seeking ceasefire talks in the face of the Saudi aerial bombardment. Bahrain's Akhbar al-Khalij daily and the Emirates' Al-Bayan accuse the Houthis of "indiscriminate shelling" of Aden residential areas.
But Arab papers elsewhere are less sanguine. Jordan's al-Ghad and Lebanon's leftist al-Safir say the Saudis have failed to halt the Houthis' advance on Aden, a development that pro-Saudi media ignore.
Pan-Arab TV channels report the fighting could draw in other players from Pakistan to Iraqi Shia militias.
Iranian media maintain their anti-Saudi stance, with official TV channels highlighting the civilian casualties of Saudi air raids and protests in various countries against Saudi policy.
A political consultant in Sanaa, Hisham al-Omeisy, told the BBC that he was sceptical that the Red Cross aid flight would go ahead.
"The Saudi coalition announced this several times before, that they will allow the ICRC to come into the airport but that hasn't actually taken place," he said.
Meanwhile, Pakistan's parliament is holding a special session to debate whether to join the Saudis and their allies.
Defence Minister Khawaja Asif, said the Saudis had asked for warships, aircraft and soldiers.
Pakistani aircraft rescued 170 people from Sanaa on Sunday. Countries including China and Egypt are planning evacuation flights, while Russia and India have already airlifted many of their nationals out of the country.
President Hadi was forced to flee Yemen two weeks ago, as the rebels advanced on Aden.
The Houthis have said their aim is to replace his government, which they accuse of being corrupt.
They are supported by troops loyal to the former President Ali Abdullah Saleh, who was ousted in the Arab Spring protests.
Saudi Arabia says the Houthis have military backing from regional rival Iran, which denies the allegation.
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An aid flight with medical supplies for Yemen has been held back because of logistical problems, as violence in the country continues.