Kidnappers have demanded ransoms of hundreds of thousands of dollars.
Six people, including a presidential guard, were sentenced to 16 years in prison on Monday for their involvement in abductions.
The guard, Arsenio Chitsotso, was a member of the elite police unit that protects President Armando Guebuza, reports the BBC's Jose Tembe from the capital, Maputo.
Two other policemen were jailed with him by a court in Maputo.
The government said police involvement in kidnappings was regrettable.
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If I were the interior minister, I would resign and get rid of the whole ministry, starting with the general police command”
Kidnappings have become common in Mozambique over the past two years - gangs initially targeted wealthy businessmen, but are now also kidnapping people from middle-class backgrounds, our reporter says.
Most kidnappings have taken place in Maputo, but the port city of Beira in central Mozambique and Nampula in the north have also been affected, he says.
Mozambique's economy has been booming in recent years, following the discovery of a major off-shore gas field.
The 13-year-old boy, named as Abdul Rashid, was found dead on Monday in the town of Dondo, about 30km (20 miles) from Beira, our correspondent says.
The boy's mother, who has not been named by the local media, accused police of conniving with the kidnappers.
The family reported his abduction to the police and immediately thereafter received a telephone call from the kidnappers who objected that "we had put the police in the middle of the deal", the mother said.
"They ended up assassinating the child," she said.
The kidnappers had originally demanded $1m (£620,000) in ransom, but lowered it to about $30,000 in negotiations, the mother said.
"We did not have this amount either. We had to pawn some of our assets, including tricycles and houses to be able to secure over $30,000 in local currency and they agreed," she is quoted by local media as saying.
"Our mistake was to contact the [police]."
Interior Minister Alberto Mondlane has promised to look into the family's allegations, our reporter says.
On Tuesday, he announced that he sacked the police criminal investigations director in Maputo, Januario Cumbana, amid a growing public outcry over the abductions.
Religious leaders and civil society groups have called a protest march for Thursday in Maputo to demand an end to the kidnappings and conflict between the army and the Renamo movement, which pulled out of a 1992 peace accord with the government last week.
Mozambican Human Rights League head Alice Mabota said the government had failed to tackle crime such as as kidnappings and drug-dealing.
"If I were the interior minister, I would resign and get rid of the whole ministry, starting with the general police command," she said.
"The problem of drugs, weapons, kidnapping - this is a very serious issue."
Rafael Anita Langa
This petition will be sent to the African, Mozambique, UN, USA and European leaders, calling them to take action to end this ongoing tragic conflict.