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Try to use your free time for reading.
Literal
Free-time [object-を] reading [for-に] use-[in-order-to-ように] endeavor-[soft-imperative-なさい].
A tidy stack of three structural pieces: the purpose particle に on 読書 ('for reading'), the ~ように clause marking the goal of the endeavor, and ~なさい as a soft imperative. 努める ('to endeavor, make an effort') takes an に-marked goal, so here it's literally 'endeavor toward [the state of] using your free time for reading.' This kind of layered purpose-framing is common in written advice.