His game is unremunerative (3)

I believe the answer is:

mug

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'his game' is the definition.
'mug' can be an answer for 'game' (thesaurus). I'm not sure about the 'his' bit.

I don't understand the remainder of the clue.

Can you help me to learn more?

(Other definitions for mug that I've seen before include "concentrate on studying" , "person's face" , "Fool - face - vessel" , "Slangy victim, by gum" , "Large cup without a saucer" .)

I've seen this clue in the Evening Standard.

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