"I saw the Ethiopian soldiers shouting at this car to stop, then it exploded," said a local resident, Abdi Hassan.
Ethiopian troops ran from the scene, Hassan said, adding that he thought there were few casualties.
Nearly half a million people have fled Mogadishu by foot, donkey and vehicle during the recent fighting and thousands are sleeping under trees or in the open in surrounding towns and villages like Afgooye.
Ethiopian tanks have pounded positions in northern and southern Mogadishu in a bid to weaken fighters.
At least three people were killed and six others wounded when a mortar crashed into a concrete building sheltering about 20 people in Tawfiq in southern Mogadishu, witnesses said.
Mutilated bodies lay rotting in the streets in northern suburbs on Tuesday, with fighting blocking access for aid workers.
Yusuf al-Azhari, a Somali political adviser, told Al Jazeera that the government evicted what he called "terrorist groups" from one of the two districts in Mogadishu.
"Mogadishu is divided into 16 districts. Two of the districts were occupied by the so-called union courts terrorists, who are supported by al-Qaeda and terrorist groups around the world.
"The government is now fighting the group to evict them from those two districts. The government forces have already succeeded in evicting them from one place."
Al-Azhari defended the presence of Ethiopian forces in Somalia, saying there are there at the invitation of the Somali government.
Ibrahim Addou, secretary of foreign affairs at the Islamic Courts Union, said the group had no relations with al-Qaeda.
"These accusations are cheap terms used for fund-raising purposes by Ethiopia. Somalia is under the Ethiopian occupation, which is being resisted by the Somali people."
The most sustained fighting in Mogadishu since the Somali-Ethiopian force defeated the Union of Islamic Courts over the New Year has killed 293 people, local officials said.
Hospitals have been overwhelmed, with doctors appealing for extra medical supplies.
Ban Ki-moon, the UN secretary-general, on Monday urged an end to the battles between fighters of the Union of Islamic Courts, some of whom are allegedly linked to al-Qaeda, and the Somali government-allied Ethiopian troops.
The US, which has backed Ethiopia's involvement in Somalia, urged all sides on Monday to agree a ceasefire, expressing concern over a growing humanitarian crisis.