"I think that it is important in the great institutions, whether it is the Army or whether it is political parties, we have had to root out attitudes... and that has to go right across the institutions."
Mr Clegg told Sky News: "He shouldn't have used these words. It will have caused considerable offence and it has obviously caused him a considerable amount of embarrassment.
"He has apologised and we should move on."
Asked if he would sack a member of his own frontbench team who used phrases like that uttered by the prince, Mr Clegg replied: "Almost certainly I would have to. Yes."
A statement from St James's Palace said: "Prince Harry fully understands how offensive this term can be, and is extremely sorry for any offence his words might cause.
"However, on this occasion three years ago, Prince Harry used the term without any malice and as a nickname about a highly popular member of his platoon.
"There is no question that Prince Harry was in any way seeking to insult his friend."
Universities and Skills Secretary John Denham said: "This sort of language can be seen as offensive, is offensive, is gradually going out of use in our society, and he's apologised for it.
"I think everybody recognises this sort of language isn't generally used these days."
A Ministry of Defence spokeswoman said: "We are not aware of any complaint having been made by the individual. Bullying and racism are not endemic in the armed forces."
A spokeswoman for the Equality and Human Rights Commission said: "These appear to be disturbing allegations and we will be asking the MoD to see the evidence, share that evidence with us and their plans for dealing with it.
"We will then consider what further action might be necessary."
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