Cleveland’s Museum of Natural History conducted the new hunt which, it was hoped, would provide the first complete with movable jaws.
Предложение не имеет смысла без контекста, а вот если прочесть предложения, которые идут перед ним, сразу становится ясно о первом ЧЁМ идет речь:
Giant earth-moving machines would cut through a formation of worldwide fame, the Cleveland shale. For more than a century it had been known as a rich source of fossil fish from the Devonian period. Specimens, collected where rivers had cut through the shale, were prized possessions of the British in New York and other centres.