"The rebels fired mortars at the airport and the presidential palace when the president arrived and the government forces responded," Mohamed Ali, a senior police official, told the AFP news agency.
"Eight civilians were killed in the market and 17 others taken to hospital," Ali Muse, the head of Mogadishu's ambulance service, said.
Clinton promised to increase arms to war-battered Somalia and further support for African peacekeepers guarding the area around the presidential palace in Mogadishu.
She said the US has provided nearly $150m to the African Mission in Somalia in the past two years and planned additional funds in the coming months.
Somalia has been mired by unrest since 1991 and is currently in the grip of a fierce violence, with opposition fighters waging a campaign to topple the African Union-backed government.