Crossword Genius

One's associated with a thinly-stretched body that's short of member (7,3)

Ross

I believe the answer is:

sausage dog

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'one's' is the definition.
Although both the answer and definition are singular nouns, I don't understand how they can define each other.

I cannot understand how the remainder of the clue works.

Can you help me to learn more?

(Other definitions for sausage dog that I've seen before include "Dachshund (colloq.)" , "At a stretch, a pet" .)

I've seen this clue in The Telegraph.
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