Smiled with scorn at Sergeant Major, vexed (7)

I believe the answer is:

smirked

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'smiled' is the definition.
(smirking is a kind of smiling)

'at sergeant major vexed' is the wordplay.
'at' means one lot of letters go next to another.
'sergeant major' becomes 'SM' (abbreviation for sergeant major).
'vexed' becomes 'irked' (I've seen this in another clue).
'sm'+'irked' is 'SMIRKED'.

'with scorn' is the link.
I am not very happy about this link. Some or all of it may be part of another bit of the clue.

Can you help me to learn more?

I've seen this clue in The Irish Times.

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