New inn dominated by duke’s small terrier (6,7)
I believe the answer is:
dandie dinmont
'small terrier' is the definition.
(type of terrier)
'new inn dominated by duke's' is the wordplay.
'new' indicates an anagram.
'by' means one lot of letters go next to another.
'duke' becomes 'd' (abbreviation).
'inn'+'dominated'='inndominated'
'inndominated' with letters rearranged gives 'andiedinmont'.
'andiedinmont' put after 'd' is 'DANDIE DINMONT'.
(Other definitions for dandie dinmont that I've seen before include "Dog" , "Small terrier with drooping ears" .)