Learn Russian: Писатель не скрывает намерения влиять на общество. - The writer doesn't hide the intention to influence society.

Because the verb скрывает is negated by не. With negative transitive verbs, the direct object often takes the Genitive case instead of the Accusative. This Genitive of negation is especially common with abstract nouns like намерение (intention).

Could we use the Accusative намерение instead of the Genitive намерения in this sentence?

Yes, but the Genitive is much more natural here. Using the Genitive (намерения) emphasizes the negation and is standard for abstract concepts. Using the Accusative (намерение) would make the intention sound like one highly specific, concrete item being hidden.

Does the verb влиять (to influence) always require the preposition на?

Yes! While in English you influence something directly, Russian requires влиять на followed by the Accusative case. Here, общество (society) is a neuter inanimate noun, so its Accusative form is identical to its Nominative form.

Why use the imperfective verb влиять instead of the perfective повлиять?

The writer's intention is to have an ongoing, continuous impact on society rather than to achieve a single, finished result. Because the focus is on a general, continuous process, the imperfective aspect is used.