Does secret in this context:
It's a story they've kept secret for a decade.
play the role of an adverb? If so why does no dictionary defines it as such?
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Think of a similar sentence:
They've kept a story secret for a decade.
The grammar here is clearly explained by the dictionary, just under "keep": the verb is used "with object and complement". The object is "a story" and "secret" is an object complement because it describes the object. An object complement can be an adjective, a noun, or a phrase that functions like an adjective or noun. Here "secret" is an adjective because it lacks an article ("a secret" is what it would be if it was a noun).
Why your sentence is ordered differently than mine is because it uses an it-cleft.
It's an adjective here.
Compare the following, which have the same structure:
First definition from OED:
secret A. adj. 1. Kept from knowledge or observation; hidden, concealed. a. Predicatively (esp. in to keep secret): Kept from public knowledge, or from the knowledge of persons specified; not allowed to be known, or only by selected persons.