Correct construction 1: The local government built the the school that was destroyed by the earthquake.
In this construction (simple past tense for both clauses), writer lays emphasis on 'built' and might discuss the the building process later on. The latter clause'that was destroyed by the earthquake' just serves to identify the school.
Correct construction 2: The local government had built the school that was destroyed by the earthquake.
In this construction (past perfect tense f/b simple past tense), writer goes back in time for a moment to refer to the building process by saying 'had built' and may lay emphasis on destruction (mentioned in simple past) and describe it further.
Both the constructions are correct.
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