Most significant time to receive care of cat (5,4)
I believe the answer is:
maine coon
'cat' is the definition.
(I know that Maine coon is a type of cat)
'most significant time to receive care of' is the wordplay.
'most significant' becomes 'main' (I've seen this in another clue).
'time' becomes 'eon' (eon is a kind of time).
'to receive' means one lot of letters goes inside another.
'care of' becomes 'co' (c/o can mean 'care of' when addressing mail).
'eon' going around 'co' is 'ecoon'.
'main'+'ecoon'='MAINE COON'
(Other definitions for maine coon that I've seen before include "Unusually big cat" , "Very large breed of cat" .)