Sooty perhaps having obligation to hold Sweep but not cry (5)
I believe the answer is:
dusty
'sooty perhaps' is the definition.
(soot is dusty or powdery)
I can't explain the remainder of the clue.
Can you help me to learn more?
(Other definitions for dusty that I've seen before include "Covered in fine particles" , "bad?" , "Dull (like a miller?)" , "In need of cleaning" , "Ms Springfield" .)