Although the internationally-recognized government established itself in Aden, President Hadi lives in exile in Saudi Arabia.
"The suicide assailant died on the spot after detonating his explosives-laden auto at a military checkpoint in Nokhan area of Shabwa province, killing 15 soldiers", the local government official told Xinhua news agency.
Aden is meant to be the "interim" capital of the pro-Saudi Hadi government, but is also the historical capital of South Yemen.
Security sources told the Associated Press that Saudi guards had stopped the separatists from entering the palace, and that the prime minister and several ministers were preparing to flee the city imminently for Riyadh.
After the separatists seized the government headquarters on Sunday, Prime Minister Ahmed bin Dagher denounced a "coup.in Aden against legitimacy and the country's unity".
Violence first erupted on Sunday when a deadline issued by the separatists for the government to resign expired.More news: Bruno Mars wins best album, song at 2018 Grammy Awards
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With reference to Yemen and what is happening in the interim capital of (Aden), the Saudi cabinet called for calm and restraint as well as adherence to the language of calm dialogue.
Both Saleh forces and the Houthi rebels are battling to capture the capital city.
Southern separatists exchanged fire with Hadi's forces in the districts of Khormaksar and Dar Saad.
He urged the Saudi-led coalition to intervene in its defence.
The clashes have sparked fears of a repeat of the 1986 South Yemen civil war, a failed socialist coup which killed thousands in just six days and helped pave the way for the 1991 unification of South and North Yemen.
They also seized two military camps near Aden worldwide airport which remained closed for the second day, the sources said.
British defence officials have been advising the Saudi-led coalition on how to conduct its controversial bombing campaign inside Yemen, although it says it is not involved in targeting decisions. Martin Griffiths, a former British diplomat and mediator with specialist Yemen knowledge, is being mentioned as his possible successor. Since then, more than 9,245 people have been killed and 3 million displaced, according to the UN.