Greeting for a dog, not one that’s barking (4,9)
I believe the answer is:
good afternoon
'greeting' is the definition.
(good afternoon is a kind of greeting)
'for a dog not one that's barking' is the wordplay.
'that's barking' indicates anagramming the letters (UK slang for mad).
'for'+'a'+'dog'+'not'+'one'='foradognotone'
'foradognotone' anagrammed gives 'GOOD AFTERNOON'.
(Other definitions for good afternoon that I've seen before include "Words of salutation" , "PM greeting" .)