Time dull routine's cut by Bob Hope (5)
I believe the answer is:
trust
'hope' is the definition.
(synonyms)
'time dull routine's cut by bob' is the wordplay.
'time' becomes 't' (abbreviation).
'dull routine' becomes 'rut' (I've seen this before).
'cut by' means one lot of letters goes inside another.
'bob' becomes 's' (abbreviation for shilling - 'bob' before 1971).
'rut' placed around 's' is 'rust'.
't'+'rust'='TRUST'
(Other definitions for trust that I've seen before include "Firm belief (in another)" , "bank on" , "Have faith in, regard as reliable" , "Fidelity" , "Hope, expect" .)