Maintain important-sounding little dog? (4,2)
I believe the answer is:
keep up
'maintain' is the definition.
(I've seen this before)
I can't explain the remainder of the clue.
(Other definitions for keep up that I've seen before include "' - - with the Joneses'" , "Stick with" , "Don't dawdle!" , "Stay in contention" , "Maintain pace with, don't fall behind (4.2)" .)