Big cat identified by secretary without hesitation (4)
I believe the answer is:
puma
'big cat' is the definition.
(I've seen this before)
'secretary without hesitation' is the wordplay.
'secretary' becomes 'PA' (abbreviation for personal assistant).
'without' indicates putting letters inside ('without' can be similar in meaning to 'outside').
'hesitation' becomes 'um' (sound made when hesitating).
'pa' going around 'um' is 'PUMA'.
'identified by' acts as a link.
(Other definitions for puma that I've seen before include "Panther" , "Big cat" , "Large wild American feline" , "South American cat" , "South American carnivore" .)