They're naturally depressed, being terribly sad about the French (5)
I believe the answer is:
dales
'they're naturally depressed' is the definition.
The definition suggests an adverb but the answer is not.
'terribly sad about the french' is the wordplay.
'terribly' is an anagram indicator.
'about' is an insertion indicator.
'the french' becomes 'le' ('the' in French).
'sad' is an anagram of 'das'.
'das' enclosing 'le' is 'DALES'.
'being' is the link.
Can you help me to learn more?
(Other definitions for dales that I've seen before include "Open river valleys in hilly areas" , "Yorkshire valleys" , "(North England) valleys" , "Leads into poetic valleys" .)