Smudge
late Middle English (as a verb in the sense ‘soil, stain’): of unknown origin. The noun dates from the late 18th century.
wiktionary
(This etymology is missing or incomplete. Please add to it, or discuss it at the Etymology scriptorium. Particularly: “Chambers 1908: "Swedish smuts dirt, Danish smuds smut, German Schmutz"”)
From Middle English smogen.
etymonline
smudge (v.)
early 15c., smogen "to soil, stain, blacken," of obscure origin. Meaning "to rub out or in" is by 1865. Related: Smudged; smudging. The noun meaning "a dirty mark or stain, spot, smear" is attested by 1768, from the verb.
The smudge meaning "make a smoky fire" is by 1860, also of unknown origin, but perhaps related. According to OED now dialectal and North American. OED also gives it in an earlier, obsolete sense of "to cure (herring) by smoking" (1590s).
The related noun smudge is attested by 1767 as "a suffocating smoke" (to repel mosquitoes, etc.); from 1806 as "heap of combustibles ignited and emitting dense smoke." Hence smudge-pot (1903). Smudge-stick as a Native American (Crow tribe) artifact is by 1908
