Barking runt dog is to run away (4,4)
I believe the answer is:
turn tail
'is to run away' is the definition.
(turn tail can mean to run away)
'barking runt dog' is the wordplay.
'barking' indicates anagramming the letters ('barking' can informally mean mad).
'dog' becomes 'tail' (synonyms).
'runt' with letters rearranged gives 'turn'.
'turn'+'tail'='TURN TAIL'
(Other definitions for turn tail that I've seen before include "Run away, deflecting rear appendage" , "retreat" , "Prepare to flee" , "Spin round and run off" , "Take to one's heels" .)