Bono said things went "badly wrong" with the charity and that he was "deeply sorry" for the situation.
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Smith also wrote that one staffer alleged her manager made "sexist and suggestive comments" about her to a government official from a different country - and then demoted her for not sleeping with the official.
The accusations first surfaced publicly when a former ONE employee from the Johannesburg office, Idriss Ali Nassah, spoke out in November on social media about the abuse accusations against the Hollywood producer Harvey Weinstein and hinted at problems at the charity. While another, as The Guardian reports, alleges that she was demoted for refusing the sexual advances of a foreign government official.
The poverty-fighting advocacy group founded by Bono and his U2 band mates is acknowledging "an institutional failure" after an internal probe uncovered an pattern of abuse and misconduct between 2011 and 2015 in the organization's office in Johannesburg, South Africa.
The Mail on Sunday said the charity admitted the issues hours after it had sent One a long list of allegations at the end of its investigation.
The charity, which aims to tackle poverty and disease in Africa, is facing legal action by former employees.
Ms. Smith said that staff members were "called names" and confirmed that some had accused a manager of demanding that they carry out domestic tasks at her home.
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The company has also discovered that its African wing was not paying taxes between 2010 and 2015, but was instead a "non-resident taxpayer".
These claims, Smith admits they are still unable to corroborate, however she noted, "We do not discount any allegation - we investigate them and will continue to do so should others arise".
She said the investigation showed "institutional failure" and that she had apologised to the former employees who agreed to speak to her.
Smith, who joined the organization in March 2017, said One Campaign learned about the allegations after the employees, who had all since left, shared their grievances on social media in November of past year.
Prominent members of the charity's board include former Prime Minister David Cameron and Sheryl Sandberg, Facebook's chief operating officer.
"My team and I heard concerns about low morale and poor management in this office but nothing along the lines of what emerged recently".
Bono said that although the allegations focus on one individual, "the head office failed to protect those employees and I need to take some responsibility for that".