Aden is the seat of power for Hadi, who has been residing in Saudi Arabia since the rebels took over the capital of Sanaa in 2014.
Another force in the anti-Huthi coalition - the UAE-trained Security Belt Force - has since Wednesday been battling loyalists in Aden, the temporary base of Hadi's government. "Wounded people dying as checkpoints prevent them reaching clinics", the ICRC said in a tweet.
On Sunday, the Red Cross urged the warring parties to sit down and negotiate the future of the country to avoid further bloodshed.
Mohammed bin Salman, the Saudi crown prince, has thrown his weight behind the Yemeni government as it battles against a separatist group backed by Saudi Arabia's allies in the United Arab Emirates (UAE).
It is also willing to attend a meeting called by Saudi Arabia, the Southern Transitional Council leader Aidaroos al-Zubaidi said in televised comments to Aden-based TV channel AIC.
The Western-backed, Sunni Muslim coalition has fractured after more than four years fighting on behalf of Hadi's ousted government against the Iran-aligned Houthi movement that controls the capital Sanaa and most populous areas.
The war has revived old strains between north and south Yemen, formerly separate countries that united into a single state in 1990.
The latest events are not the first example of southern separatists opposing forces loyal to Hadi.
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News agencies reported on Sunday about Yemeni's separatists beginning to backslide from positions seized from Abdul Mansourhadi's government in Aden.
Separatist commanders accused the Al-Islah Islamist party of killing one of its commanders and "infiltrating" the Hadi government, which is politically and financially supported by Saudi Arabia.
In the last few days, dozens of civilians and fighters were killed in the fighting between the separatists and government forces.
After the takeover of the sites in Aden by the Emirati-backed forces, Hadi's regime said it held the UAE and militants backed by it responsible for "the coup" in the city.
The UAE has been under pressure for supporting the separatist militias in southern Yemen, which are part of the so-called Security Belt.
The internationally recognised government's interior minister, however, blamed the presidential office and ally Saudi for remaining "silent" while acknowledging the UAE's victory in the south.
Following the airstrike, the STC forces, which had taken over the government's military bases and the presidential palace, announced its withdrawal on the Saudi-led coaition's request.
The force, known as the Security Belt, overran three military barracks belonging to unionist forces on Saturday and were surrounding the presidential palace, sources close to the force said.
He stressed the need to "preserve unity and stability" in Yemen which, according to the International Crisis Group think tank, is facing the risk of a "civil war within a civil war".