It’s an Algonquin word originally, seemingly not related to “punk.”
Here’s the OED entry:
Pronunciation: Brit. /?p??ki/, U.S. /?p??ki/ Forms: 17?" punkie, 18
pungie, 18?" punkey, 18?" punky, 19?" ponki, 19?" punk. Etymology:
New York Dutch *punkje Munsee Delaware p??nkw?s ( Proto-Algonquian
*penkw- dust, ashes + -ehs-, diminutive suffix), with alteration of
the ending after Dutch -je, diminutive suffix. U.S. regional (chiefly
A biting midge of the family Ceratopogonidae; a no-see-um.
1769 R. Smith Tour Four Great Rivers (1906) 42 We begin to be teazed
with Muscetoes and little Gnats called here [in New York] Punkies.
1840 Knickerbocker Mag. 16 270 The tortures…inflicted by the gnats
(sand-flies, punkies, brulos, for they bear all these appellations).
1876 Forest & Stream 13 July 368/2 Hands tingling from punkie bites.
1903 S. E. White Forest 154 The midge, again, or punkie, or
?no-see-'um?, just as you please, swarms down upon you suddenly.
1933 F. H. Cheley Camping Out 423 The ?punkeys? and ?midgets? can
outstrip them [sc. mosquitoes] for ferocity and the painful
character of the wound which they inflict.
2003 C. Palmer & P. Stoks Loved One 39 The chill of the night kept
the mosquitoes and punkies at bay.