Periodically miss old cover of Adam Faith (5)
I believe the answer is:
islam
'faith' is the definition.
(Islam is an example)
'periodically miss old cover of adam' is the wordplay.
'periodically' means one should take alternating letters (letters must be selected 'every so often').
'cover of' means to remove the middle letters (only the letters which cover the word).
'adam' with its middle removed is 'am'.
'miss'+'old'='missold'
The alternate letters of 'missold' are 'isl'.
'isl'+'am'='ISLAM'
(Other definitions for islam that I've seen before include "Religion of the Koran" , "Religious faith of Muslims" , "Religion based on Muhammad's teachings" , "Religion based on teachings of Muhammad" , "Major world religion" .)