Animal with sensitivity not entirely laid back (3)
I believe the answer is:
cat
'animal' is the definition.
(cats are animals)
'sensitivity not entirely laid back' is the wordplay.
'sensitivity' becomes 'tact' (being tactful is being sensitive).
'not entirely' means to remove the last letter (not all of the word).
'laid back' says the letters should be written in reverse.
'tact' with its last letter taken away is 'tac'.
'tac' in reverse letter order is 'CAT'.
'with' is the link.
(Other definitions for cat that I've seen before include "A tom" , "one's pyjamas? They're an excellent thing" , "Domestic animal" , "Carnivore; whip" , "Animal (domesticated or wild)" .)