Awfully big lie — one fired with yen for fitness (11)
I believe the answer is:
eligibility
'fitness' is the definition.
(eligibility is a kind of fitness)
'awfully big lie one fired with yen' is the wordplay.
'awfully' is an anagram indicator.
'one' becomes 'i' (Roman numeral**).
'fired' becomes 'lit' (set on fire).
'with' says to put letters next to each other.
'yen' becomes 'y'.
'big'+'lie'='biglie'
'biglie' is an anagram of 'eligib'.
'eligib'+'i'+'lit'+'y'='ELIGIBILITY'
'for' is the link.
(Other definitions for eligibility that I've seen before include "Fitness to be chosen" , "Suitableness" .)