A lack of training? That's not gonna happen! (4)
I believe the answer is:
nope
'not gonna happen' is the definition.
(nope can informally mean no)
'a lack of training?' is the wordplay.
'a lack of' becomes 'no' (lack of money is no money).
'training?' becomes 'PE' (Physical Education).
'no'+'pe'='NOPE'
'that's' acts as a link.
(Other definitions for nope that I've seen before include "Not on your life!" , "Emphatically not!" , "Certainly not" , "Negative response" , "I'm afraid not" .)