It was the second attack in days claimed by IS in the city of 200,000 people which was recaptured by government forces from the rival jihadists of Al-Qaeda with US backing.
The suicide bomber killed at least 31 police recruits on the southwestern outskirts of the city, which is the capital of Hadramawt province, medics said.
The bomber detonated an explosives belt as he joined a line of men at a police recruitment centre, a provincial official said.
Hadramawt's security chief, General Mubarak al-Oubthani, who was at the recruitment centre at the time of the attack but was not hurt, was the target of a second bombing when he headed to the centre of Mukalla afterwards, a security official said.
Also Read
The bomb went off as Oubthani walked out of his office killing six of his guards but leaving him with only minor injuries, the official said.
An IS statement posted online claimed the suicide attack, the second but rare intervention by the jihadist group in an area known as a stronghold of rival Al-Qaeda.
On Thursday, 15 Yemeni troops were killed in jihadist attacks on army positions outside Mukalla. IS said one of its militants blew up a vehicle packed with explosives in an army base in Khalf district on the city's eastern outskirts.
The attacks included a suicide bombing that targeted the residence of the commander of Hadramawt's second military region, General Faraj Salmeen, but he escaped unharmed, officials said.
The general boasted on Friday that his forces had captured some 250 Al-Qaeda members since they retook Mukalla and nearby coastal towns, including its commander for the city of Shihr, some 60 kilometres to the east.
The Pentagon revealed last week that a "very small number" of US military personnel had also been deployed around Mukalla in support of the operation.
The US Navy has several ships nearby, including an amphibious assault vessel, USS Boxer, and two destroyers.
The offensive against Al-Qaeda comes amid a truce and peace talks between the government and Iran-backed rebels it has been fighting with support from a Saudi-led coalition since March last year.