Period of lost hope encompassing century (5)
I believe the answer is:
epoch
'period' is the definition.
(epoch is a kind of period)
'lost hope encompassing century' is the wordplay.
'lost' is an anagram indicator.
'encompassing' is an insertion indicator.
'century' becomes 'c' (common abbreviation).
'hope' anagrammed gives 'epoh'.
'epoh' going around 'c' is 'EPOCH'.
'of' is the link.
(Other definitions for epoch that I've seen before include "Major division or period of history" , "Unit of geological time" , "Making this date" , "Period of time in history" , "Geological time period" .)