Go without fixed purpose in pilgrimage site reportedly (4)
I believe the answer is:
roam
'go without fixed purpose' is the definition.
(to roam around is to travel without fixed purpose)
'pilgrimage site reportedly' is the wordplay.
'pilgrimage site' becomes 'Rome' (Catholics may go on a pilgrimage to Rome).
'reportedly' indicates a 'sounds like' (homophone) clue.
'rome' is a homophone of 'ROAM'.
'in' acts as a link.
(Other definitions for roam that I've seen before include "Rove or ramble" , "Travel widely" , "Meander" , "Walk without a specific destination" , "Ramble without purpose" .)