Better put on boring headgear (4,3)
I believe the answer is:
flat cap
'headgear' is the definition.
(flat cap is a kind of headgear)
'better put on boring' is the wordplay.
'better' becomes 'cap' (both can mean to beat somebody else's attempt).
'put on' says to put letters next to each other (some letters go on or after others).
'boring' becomes 'flat' (flat can mean dull or lifeless).
'cap' put after 'flat' is 'FLAT-CAP'.
(Other definitions for flat cap that I've seen before include "working-class symbol" , "> lid" , "Headgear (of the working class?)" , "Type of hat" , "Style of headgear" .)