Dog Zeus initially confined to his hut that's barking (4,3)
I believe the answer is:
shih tzu
'dog' is the definition.
(Shih-Tzu is an example)
'zeus initially confined to his hut that's barking' is the wordplay.
'initially' suggests taking the first letters.
'confined to' is an insertion indicator.
'that's barking' indicates anagramming the letters ('barking' can informally mean mad).
The first letter of 'zeus' is 'z'.
'his'+'hut'='hishut'
'hishut' anagrammed gives 'shihtu'.
'z' going into 'shihtu' is 'SHIH-TZU'.
(Another definition for shih tzu that I've seen is " Toy dog with long silky hair".)