Majority think again briefly when cutting caper (13)
I believe the answer is:
preponderance
'majority' is the definition.
(thesaurus)
'think again briefly when cutting caper' is the wordplay.
'think again' becomes 're-ponder' (ponder again).
'briefly' means to remove the last letter.
'when cutting' means one lot of letters goes inside another (some letters cut their way into a word).
'caper' becomes 'prance' (both can mean to skip or dance around).
'reponder' with its last letter taken off is 'reponde'.
'reponde' put within 'prance' is 'PREPONDERANCE'.
(Other definitions for preponderance that I've seen before include "The lion's share" , "Superiority in numbers" , "Prevalence" , "In majority of" , "Greater quantity" .)