Difficulty dropping duke's race horse equipment (7)

I believe the answer is:

harness

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'equipment' is the definition.
(harness is a kind of equipment)

I don't understand the remainder of the clue.

(Other definitions for harness that I've seen before include "Set of straps for horse etc" , "Control and use (resources)" , "Set of straps for horse or child" , "Control - some tack" , "Control; trappings" .)

I've seen this clue in The Times.

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