Get their mitts on the nest eggs! (6)
I believe the answer is:
Here is my best explanation:
'eggs' is the definition.
(I've seen this before)
I don't understand how the remainder of the clue works.
(Other definitions for clutch that I've seen before include "(Try to) grasp tightly", "Result of laying", "result", "Tightly grasp, eggs", "Grasp - eggs laid in nest", "Grip part of the car-engine", "Brood of chickens", "Grasp - eggs - car pedal", "Take hold of, grab" and "Get hold of".)