Heavy demand, dash it! (3)

I believe the answer is:

run

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'heavy demand dash' is the definition.
'run' can be an answer for 'dash' (thesaurus). I am not certain of the 'heavy demand' bit.

I can't explain the rest of the clue.

Can you help me to learn more?

(Other definitions for run that I've seen before include "Flow; smuggle (goods)" , "stand" , "Proceed quickly" , "Organise - series" , "Discharge" .)

I've seen this clue in the Evening Standard.

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