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She's acting on some selfish motive.
Literal
She [topic-は] some selfish motive [with-で] is-acting.
何か + noun ('some N') is the indefinite counterpart to 何 in questions — 'what' becomes 'something' when paired with か. Here it implies 'a motive of some kind, though I'm not sure exactly what.' 利己的 ('self-interested,' from 利 'benefit' + 己 'self' + 的 adjectival suffix) belongs to a higher, slightly philosophical register; in casual speech 自分勝手 ('selfish, self-centered') sounds more natural for the same idea.