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The allegations, which purportedly involve powerful and famous figures of that era including politicians, made headlines after an opposition lawmaker raised them in parliament.
They have resurfaced at a time when the British authorities are investigating and trying to prosecute celebrities and other well-known figures in public life over other unrelated historic allegations of sexual misconduct.
The opposition Labour Party has called for an "overarching review" into the child abuse allegations, accusing Cameron's Conservative-led government of not doing enough and of failing to grasp the matter's gravity.
"Given the extent of concern about this, Theresa May (the Home Secretary) should not simply be leaving it to officials and to the prime minister to resolve," Yvette Cooper, Labour's spokeswoman for home affairs, said in a statement.
"She needs to make sure there is a process people can feel confident in - to get truth and justice, but also to protect children in future."
The way the public perceives Cameron's government's handling of the allegations is important for the British leader, who is up for re-election next year.
INDEPENDENT LEGAL FIGURE
One of the country's top civil servants, Mark Sedwill, wrote to Cameron on Saturday to say he was appointing a "senior independent legal figure" to judge whether the conclusions of an internal review into the matter last year remained sound. That person would be appointed "within the next week", he said.
Sedwill said separately on Saturday in another letter that the review last year had concluded that Britain's Home Office (interior ministry) had passed nine allegations about child abuse onto the appropriate authorities at the time.
He said the review had uncovered a further four pieces of information which had not been previously disclosed which had since been passed to the police.
In comments which prompted some politicians to talk of a possible cover-up, he said that 114 "potentially relevant files" had been destroyed, were missing or could not be found.
Michael Gove, Cameron's education minister, said on Sunday that whilst he did think it was important the allegations were looked at he did not think a full-scale inquiry was necessary.
"It's also important, I think, to emphasise that many of the allegations that are being made are historic and that what we do now ... in order to keep children safer is immeasurably better and stronger than was the case, you know twenty or thirty years ago," Gove told BBC TV.
"No", he said when asked if the government should undertake a public inquiry.
Norman Tebbit, a former Conservative minister, said he thought there "may well" have been a political cover-up in the 1980s, but said: "But it was almost unconscious. It was the thing people did at that time." However, David Mellor, another former Conservative minister, said he thought the matter had been exaggerated and that a "witch hunt" was underway.
In 2012, police said Jimmy Savile, one of Britain's best-known TV presenters in the 1970s and 1980s, had sexually abused hundreds of victims, mainly youngsters, at hospitals and BBC premises over six decades until his death aged 84 in 2011.
On Friday, entertainer Rolf Harris, a household name in his native Australia and his adopted home Britain, was jailed for almost six years for repeatedly abusing young girls over decades as a host of children's television.