Essential element of Depression: onset of hunger (4)
I believe the answer is:
pith
'essential element' is the definition.
(pith can mean something's core or essence)
'depression onset of hunger' is the wordplay.
'depression' becomes 'pit' (pit is a kind of depression).
'onset of' suggests taking the first letters.
The initial letter of 'hunger' is 'h'.
'pit'+'h'='PITH'
'of' acts as a link.
(Other definitions for pith that I've seen before include "Essential substance" , "Citrus rind lining" , "Marrow (of plant)" , "The white part of the orange" , "Orange product" .)