Many people were injured when the boy struck at the market in Paktika province. A local official, a woman and two men died.
The suicide bomber was a 12-year-old boy, said a statement from the governor's office.
In Ghazni city, a man opened fire at a police checkpoint, killing two policemen and two civilians.
A bicycle bomb blast near the city's main police station injured 13 people.
On Saturday, the Taliban declared a new attack on foreign troops, Afghan security forces and government officials.
International organisations, including the UN, have warned their staff not to go out unless absolutely necessary.