Between cigarette ends, the band will fawn or grovel (6)
I believe the answer is:
cringe
'grovel' is the definition.
(synonyms)
'between cigarette ends the band will fawn' is the wordplay.
I cannot really see how this works, but
'band' could be 'ring' (a ring is a band of metal) and 'ring' is located in the answer.
This explanation may well be incorrect...
'or' acts as a link.
Can you help me to learn more?
(Other definitions for cringe that I've seen before include "Shrink back, cower" , "Shrink away in fear" , "Cower in fear or disgust" , "obeisance" , "Recoil in embarrassment" .)