A blemish or smear, especially a dark or sooty one.
There was a smudge on the paper.
Dense smoke, such as that used for fumigation.
A heap of damp combustibles partially ignited and burning slowly, placed on the windward side of a house, tent, etc. to keep off mosquitoes or other insects.
(especially in the phrase "smudge stick" = "stick of incense") A quantity of herbs used in suffumigation.