。
She was desperately trying to banish her doubts.
Literal
She [topic-は] heart [from-から] doubts [object-を] remove-try-to desperate was.
取り払う (とりはらう, 'to remove, to clear away') is a compound verb combining 取る ('take') with 払う ('sweep away'). The volitional ~おうとする structure ('trying to') expresses her effort. 懸命 (けんめい) means 'desperate, frantic' — she was throwing everything at removing these doubts. 心から marks the source: the doubts are rooted in her heart/mind.