Bachelor to race on horseback and take a refresher course (5-2)

I believe the answer is:

brush-up

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'and take a refresher course' is the definition.
I can't tell whether this defines the answer.

'bachelor to race on horseback' is the wordplay.
'bachelor' becomes 'B' (abbreviation e.g. BA=Bachelor of Arts).
'to race' becomes 'rush' ('rush' can be a synonym of 'race').
'on horseback' becomes 'up' (up can mean up on a horse).
'b'+'rush'+'up'='BRUSH-UP'

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I've seen this clue in the Sydney Morning Herald.

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