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"shaky, groggy, unstable," 1919, of unknown origin. Cf. German prefix wankel-, which has a similar sense. Or perhaps from surviving dialectal words based on Old English wancol "shaky, tottering" (see wench).
In a halting and unsteady way: We commended him on his performance and sent him wonkily tottering back onto the floor (1990s+ College students)
Redolent of policy, theory, discussion, etc: Greenberg felt that a wonkish tone had pervaded the transition (1990s+)