Animal’s limb badly beset by trouble (9)
I believe the answer is:
armadillo
'animal's' is the definition.
(armadillo is a kind of animal)
'limb badly beset by trouble' is the wordplay.
'limb' becomes 'arm' (arm is a kind of limb).
'badly' becomes 'ill' ('ill' can be adverbial as in 'ill-advised').
'beset by' indicates putting letters inside ('beset' could historically mean 'cover').
'trouble' becomes 'ado' (I've seen this before).
'ill' put within 'ado' is 'adillo'.
'arm'+'adillo'='ARMADILLO'
(Other definitions for armadillo that I've seen before include "Dollar, ami for S American animal with bony plates" , "Mammal covered with bony plates" , "Plated burrowing animal" , "American insect-eater" , "Scaly mammal" .)