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Candy's dog represents Candy. It is old, arthritic, and has little practical use. Candy muses about whether they will shoot him when he is no good. There is a little foreshadowing here as well with a similar fate for Lennie.
Lennie's puppy symbolizes himself. It personifies a friend that Lennie desperately needs yet kills it. Similarly George must kill his fiend Lennie as well. The puppy also represents innocent emotional/intillectual frailty as Lennie has as well.