Occasionally, diary close to Pepys illustrates parts of the week (4)
I believe the answer is:
days
'parts of the week' is the definition.
(a week is made up of days)
'occasionally diary close to pepys' is the wordplay.
'occasionally' indicates alternate letters.
'close to' says to take the final letters (the close/ending of).
The last letter of 'pepys' is 's'.
The alternate letters of 'diary' are 'day'.
'day'+'s'='DAYS'
'illustrates' is the link.
(Other definitions for days that I've seen before include "Periods (of week)" , "24-hour periods" , "Time period(s)" , "Periods of 24 hours" , "Twenty four hour periods" .)