This is the complete sentence:
Certain Italians who were newly cognizant of Greek and Roman cultural accomplishments initiated a classical cultural rebirth after a long period of stasis.
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In this sentence, "newly" modifies "cognizant".
It is not idiomatic for "newly" to modify "were". In fact, it is difficult to conceptualize what that would even mean.
Furthermore, the context provided by the rest of the sentence confirms this interpretation: "initiated a classical cultural rebirth after a long period of stasis." This rebirth after stasis is a consequence of their new cognizance.
Newly applies to cognizant.
Newly is used before a past participle or an adjective to indicate that a particular action is very recent, or that a particular state of affairs has very recently begun to exist.