Pet racing greyhound? (6)

I believe the answer is:

lapdog

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'pet' is the definition.
(I know that lapdog is a type of dog)

'racing greyhound?' is the wordplay.
'racing' becomes 'lap' (I can't explain this - if you can you should believe this answer much more).
'greyhound?' becomes 'dog' (greyhound is a kind of dog).
'lap'+'dog'='LAPDOG'

Can you help me to learn more?

(Other definitions for lapdog that I've seen before include "One docile and subservient" , "devoted supporter" , "Small pampered pet" , "Submissive type" , "Pampered pet - sycophantic sidekick" .)

I've seen this clue in the Evening Standard.

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