Be an eager beaver (4)
I believe the answer is:
gnaw
'be an eager beaver' is the definition.
(I've seen this in another clue)
This is all the clue.
(Other definitions for gnaw that I've seen before include "Eat at" , "Prey on one's mind" , "Bite like a rodent" , "Chew persistently at, say a bone" , "Chew at, work with teeth" .)