Pet, maybe, to fasten up a little (6)
I believe the answer is:
rabbit
'pet' is the definition.
'to fasten up a little' is the wordplay.
'to fasten' becomes 'bar' (barring is a kind of fastening).
'up' is a reversal indicator (in down clue: letters go upwards).
'a little' becomes 'bit' (both can mean a small amount of something).
'bar' reversed gives 'rab'.
'rab'+'bit'='RABBIT'
'maybe' acts as a link.
(Other definitions for rabbit that I've seen before include "one found in the tail" , "Poor player" , "bun?" , "Come up with a solution and pull one out of the hat" , "Inferior sportsman" .)