|The AU force is in Somalia to help the UN-backed government defeat insurgents|
The troops had "misused" their power over women fleeing violence and poverty, Human Rights Watch (HRW) said.
One Muslim girl, aged 15, reportedly had her headscarf ripped off before being raped.
The AU, which has some 22,000 troops fighting militant Islamists in Somalia, says it will investigate the claims.
|A destroyed car after an attack by militants at the Jilacow underground cell |
inside a national security compound in Mogadishu (31 August 2014)
Al-Shabab carries out periodic attacks in Mogadishu
'Raped girl paid'
Last year, there was a huge public outcry following claims that AU and government soldiers had gang-raped a woman in Mogadishu.
|Faduma Aden Mohamud holding a baby as she cooks samosas in her house |
in a displacement camp in the Somali capital, Mogadishu
Tens of thousands of people have been forced to flee their homes because of conflict
Most of the women who were abused lived in camps for displaced people after fleeing violence and the 2011 famine, HRW said.
"The AU can no longer turn a blind eye to the abuses on Amisom [AU Mission in Somalia] bases, as it's undermining the very credibility of the mission," said Liesl Gerntholtz, HRW's women's rights director.
No kidding! They are starting to make Al-Shabaab look good! Their actions only push Somalis toward the more radical elements of Islam defeating the very purpose of being there.
It interviewed 21 women and girls, some as young as 12, who described being raped or sexually exploited by Ugandan or Burundian soldiers in the AU force, HRW said.
Only one rape case, in which the victim was a child, is before Uganda's military court in Kampala, it added.
"Some Amisom soldiers have used humanitarian assistance, provided by the mission, to coerce vulnerable women and girls into sexual activity," HRW said.
"A number of the women and girls interviewed for this report said that they were initially approached for sex in return for money or raped while seeking medical assistance and water on the Amisom bases, particularly the Burundian contingent's base."
A Burundian soldier gave a 15-year-old girl $10 (£6) after raping her, HRW said.
"First he ripped off my hijab [headscarf] and then he attacked me," it quoted her as saying.
The girl had gone to an AU base of Burundian soldiers to collect medicine for her mother who was ill, HRW said.
Somalia has been hit by instability since the overthrow of long-serving ruler Siad Barre in 1991.
The AU force was deployed in 2007, with most of its troops coming from Uganda and Burundi.
Last week, the USA announced Al-Shabaab leader Ahmed Abdi Godane was killed by a US air strike in Somalia. Pentagon hails ‘major symbolic and operational loss’ for Islamist militant group with death of Ahmed Abdi Godane.
|Al-Shabaab ushers in new leadership after deadly US air strike|