Staff’s methodology when hosting a travelling exhibition (8)
I believe the answer is:
roadshow
'travelling exhibition' is the definition.
(a touring show)
'staff's methodology when hosting a' is the wordplay.
'staff's' becomes 'rod's' (both can mean a stick or cane).
'methodology' becomes 'how' (how something is done).
'when hosting' is an insertion indicator (the word welcomes or hosts inserted letters).
'rods'+'how'='rodshow'
'rodshow' placed around 'a' is 'ROADSHOW'.
(Other definitions for roadshow that I've seen before include "political campaign" , "travelling event" .)