Crossword Genius

Reportedly unproductive bodily vessel (4)

Ross

I believe the answer is:

vein

I'm a little stuck... Click here to teach me more about this clue!

'bodily vessel' is the definition.
'vein' can be an answer for 'vessel' (vein is a kind of vessel). I am unsure of the 'bodily' bit.

'reportedly unproductive' is the wordplay.
'reportedly' shows a homophone (sound like).
'unproductive' becomes 'vain' (similar in meaning).
'vain' sounds like 'VEIN'.

Can you help me to learn more?

(Other definitions for vein that I've seen before include "Blood-carrying vessel" , "Vessel for claret" , "Leaf rib; ore layer in rock" , "Vessel carrying blood towards the heart" , "Deposit of mineral" .)

I've seen this clue in The Times.
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