Rot's primarily engulfing the old part of dartboard (5,3)
I believe the answer is:
bulls eye
'part of dartboard' is the definition.
(centre of a dartboard)
'rot's primarily engulfing the old' is the wordplay.
'rot's' becomes 'bull's' ('bull' can be a synonym of 'rot').
'primarily' indicates taking the first letters.
'the old' becomes 'ye' (resembles a historical spelling of 'the').
The initial letter of 'engulfing' is 'e'.
'bulls'+'e'+'ye'='BULLS-EYE'
(Other definitions for bulls eye that I've seen before include "Light - gold - sweet" , "Window - sweet" , "A small round window" , "shot, right on target" , "cow passing by might catch it" .)