What is the role of "destitute" in this sentence?

The part of speech is adjective, but its function in this sentence is that of an appositive. Latin appositio means "a placing near". An appositive is a nominal phrase (with as its core either an adjective or a substantive noun) modifying a noun, but in a way that feels 'tacked on', almost like a separate praedicate. The comma and the unusual placement make it clear that this is an appositive.

Other examples:

Cleopatra left the palace, carrying a snake in a basket.

Mother of Caesarion, she had great hopes for her dynasty.

But Octavian, Caesar's adopted son, had him put to death.

Cleopatra, long dead, never had to face this dashing of her hopes.

The ordinary, non-appositive use of adjectives is called attributive (e.g. "the destitute girl will sit on the ground").

P.S. This adjective is not absolute, because it modifies she: absolute constructions do not modify any nouns, but they stand on their own as adverbial phrases. The word 'absolute' means "loose, detached" from the main praedicate and its (other) constituents.