Made heavy demands on time prior to cut (5)
I believe the answer is:
taxed
'made heavy demands on' is the definition.
(I've seen this before)
'time prior to cut' is the wordplay.
'time' becomes 't'.
'prior to' means one lot of letters go next to another.
'cut' becomes 'axed' (I've seen this before**).
't'+'axed'='TAXED'
(Other definitions for taxed that I've seen before include "Stretched" , "Heavily burdened" , "Made heavy demands on" , "Imposed duty on" , "Weighed down" .)