In the earlier attack, fighters stormed the police checkpoint in Maywand district of Kandahar province, Sadullah Khan, a police officer in neighbouring Neven district, told the Associated Press news agency.
After the attack, the fighters took two police vehicles and the officers' weapons, Khan said.
The attack in Lashkar Gah, the capital of the Taliban-dominated Helmand province, on Sunday, took place near the home of a senior police official, authorities said.
The bomber tried to enter the house of the regional police commander wearing an explosive vest, Mohammad Hussain Andiwal, the provincial police chief, said.
"He was stopped by our policemen present there. When they stopped him, he [the bomber] exploded himself and martyred one policeman."
The injured included four policemen and two children.