Distribute carelessly, it's true (5)

I believe the answer is:

strew

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'distribute' is the definition.
(I know that strew is a type of distribute)

'carelessly it's true' is the wordplay.
'carelessly' is an anagram indicator.
'true' becomes 'rew' (I can't justify this - if you can you should believe this answer much more).
'ts' with letters rearranged gives 'st'.
'st'+'rew'='STREW'

Can you help me to learn more?

(Other definitions for strew that I've seen before include "Scatter untidily" , "Sprinkle" , "Scatter, spread around" , "Scatter loosely" .)

I've seen this clue in the Evening Standard.

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