By Mohammed Mukhashaf and Sami Aboudi
ADEN, March 29 (Reuters) - Yemeni fighters loyal to the Saudi-backed President Abd-Rabbu Mansour Hadi clashed with Iranian-allied Houthi fighters on Sunday in downtown Aden, the absent leader's last major foothold in the country.
Hadi loyalists in the southern port city reported a gun battle in the central Crater district in which three people were killed, and said they recaptured the airport, which has changed hands several times in the last five days of fighting.
The Health Ministry, loyal to the Houthi fighters who control the capital, said Saudi-led air strikes had killed 35 people and wounded 88 overnight. The figures could not be independently confirmed.
The Houthi fighters, representing a Shi'ite minority that makes up around a third of Yemen's population, emerged as the most powerful force in the Arab world's poorest country last year when they captured the capital Sanaa.
Saudi Arabia has rallied Sunni Muslim Arab countries in an air campaign to support Hadi, who relocated to Aden in February and is now in Riyadh after leaving Yemen in the past week. The fighting has brought civil war to a country that was already sliding into chaos and which had been a battlefield for the secret U.S. drone war against al Qaeda.
In the eastern province of Shabwa, tribal sources said armed tribesmen were fighting a major battle with the Houthis and their army allies at a military base, killing around 30 Houthis. This also could not be independently confirmed.
Saudi-led coalition warplanes struck military targets at airports in the capital Sanaa and in Hodeida, the main Red Sea port.
In the northern city of Saada, a Houthi stronghold near the Saudi border, strikes hit Houthi military bases belonging to the militia and their ally, former President Ali Abdullah Saleh who still controls most army units.
Saleh stood down after a 2011 uprising but still wields wide influence in Yemen. He appealed on Saturday to Arab leaders meeting in Egypt to halt their four-day offensive and resume talks on political transition in Yemen, promising that neither he nor his relatives would seek the presidency.
In an apparent rebuttal, a Yemeni official said Hadi had sacked Saleh's son as ambassador to the United Arab Emirates.
Saudi Arabia's military intervention, is the latest front in its widening contest with Iran for power in the region, a proxy struggle also playing out in Syria, Iraq and Lebanon.
Iran denies accusations from Sunni Gulf rulers that it has armed the Houthis, who follow the Zaidi branch of Shi'ite Islam.
Zaidi Shi'ites led a thousand-year kingdom in Yemen until 1962. Former leader Saleh himself is a member of the sect, although he sought to crush the Houthis while in office, only allying himself with them after his downfall.
FIGHTING IN ADEN CENTER
Across the country, there were heavy clashes in seven southern and eastern provinces between the Houthis and pro-Saleh army units on the one hand, against Sunni tribesmen, pro-Hadi loyalists and armed southern separatists on the other.
Fighters loyal to Hadi said on Sunday they recaptured Aden airport after fighting which lasted all night. Heavy fighting in the area during the last week meant that foreign diplomats had to be evacuated from the city by boat, ferried by Saudi naval vessels to the Red Sea port of Jeddah on Saturday.
Witnesses in Aden's northern Dar Saad district reported seeing two tanks destroyed by Hadi loyalists battling army units who are fighting alongside the Houthis. Five members of the Shi'ite militia were killed in the suburbs, loyalists said.
In comments addressed to Arab heads of state meeting in Egypt, Saleh appealed to the Saudi-led coalition on Saturday to stop "the aggression and return to the negotiations table," saying Hadi had failed to run the country.
"Let's go to dialog and elections, and I promise you that neither I nor any of my relatives will run for the presidency."
But in public at least, Saudi Arabia and Yemeni officials loyal to Hadi appeared to give Saleh short shrift.
Saudi Arabia's King Salman told the summit that military operations would continue until their objectives were achieved.
Hadi's Foreign Minister Riyadh Yaseen said: "We are not thinking about dialog in the present time until the conditions are arranged on the ground."
In a rare move, Saudi-owned television channel Al-Arabiya broadcast a detailed account of what it said was a proposal last week to the Saudi leadership by Saleh's son Ahmed to head off military intervention by breaking with the Houthis.
It said two days before the Saudi-led campaign began, Ahmed Saleh offered in a meeting with Saudi Defense Minister Prince Mohammad bin Salman to organize a coup against the Houthis.
In return, the channel said, Saleh requested U.N. sanctions on his father be lifted, immunity be granted to him and his father, and media campaigns against his father be halted.
Al-Arabiya said Prince Mohammad rejected the proposal. "There must be a return to legitimacy in the form of President Abd-Rabbu Mansour Hadi to lead Yemen from the capital Sanaa," it quoted him as saying. (Additional reporting by Noah Browning in Dubai, Mohammed Ghobari in Cairo and Yara Bayoumy in Sharm el-Sheik; Writing by Dominic Evans; Editing by Peter Graff)