Minor branch’s special apparatus (5)

I believe the answer is:

sprig

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'minor branch's' is the definition.
'sprig' can be an answer for 'branch's' (sprig is a kind of branch). I am unsure of the 'minor' bit.

'special apparatus' is the wordplay.
'special' becomes 'sp' (abbreviation).
'apparatus' becomes 'rig' (I've seen this in another clue).
'sp'+'rig'='SPRIG'

Can you help me to learn more?

(Other definitions for sprig that I've seen before include "Shoot - twig" , "Small branch or shoot" , "Tree shoot; headless nail" , "Spray - youngster" , "Small spray of leaves or flowers" .)

I've seen this clue in The Times.

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