Smeared dirt, say, on the rise in South Dakota (7)
I believe the answer is:
smudged
'smeared' is the definition.
('smudge' can be a synonym of 'smear')
'dirt say on the rise in south dakota' is the wordplay.
'dirt' becomes 'mud' (mud is a kind of dirt).
'say' becomes 'eg' (both can mean for example).
'on the rise' says the letters should be written in reverse (in a down clue, letters go up).
'in' indicates putting letters inside.
'south dakota' becomes 'sd' (abbreviation).
'eg' in reverse letter order is 'ge'.
'mud'+'ge'='mudge'
'mudge' placed within 'sd' is 'SMUDGED'.
(Other definitions for smudged that I've seen before include "Smeared" , "Soiled or blurred through rubbing" , "Blotted" , "Made blurred" .)