Part of circle primarily haunting gateway, perhaps (4)
I believe the answer is:
'gateway perhaps' is the definition.
'part of circle primarily haunting' is the wordplay.
'part of circle' becomes 'arc' (part of the circumference of a circle).
'primarily' says to take the initial letters.
The first letter of 'haunting' is 'h'.
(Other definitions for arch that I've seen before include "Chief - part of foot" , "Prefix meaning cheif, perhaps enemy" , "Inner part of foot" , "Roguish, teasing" , "Affectedly playful or extreme" .)