Accuse falsely, according to some heartlessly (7)

I believe the answer is:

asperse

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'accuse falsely' is the definition.
(I've seen this before)

I don't understand how the remainder of the clue works.

(Other definitions for asperse that I've seen before include "Slander or calumniate" , "Spares E. only to slander him" , "See spar could slander oddly" , "Call names" , "Defame" .)

I've seen this clue in The Sunday Times and The Times.

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