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Please clean my room while I'm away.
Literal
I [of-の] absence during [in-に] room [object-を] clean-put [please-下さい].
留守中 ('during absence') is a fixed temporal noun meaning 'while one is away from home.' 掃除しておく uses ~ておく, the preparatory auxiliary meaning 'do [in advance for future use].' So this is 'have the cleaning done ahead of time.' ください politely requests. Note the omitted speaker — 僕の starts the sentence without 僕が or 僕は.