Check on what sounds like costly animal (8)
I believe the answer is:
reindeer
'animal' is the definition.
(thesaurus)
'check on what sounds like costly' is the wordplay.
'check' becomes 'rein' (both can mean to control or limit).
'on' means one lot of letters go next to another.
'what sounds like' shows a homophone (sound like).
'costly' becomes 'dear' ('dear' can be a synonym of 'costly').
'dear' is a homophone of 'deer'.
'rein'+'deer'='REINDEER'
(Other definitions for reindeer that I've seen before include "Beasts of the Arctic" , "Christmas card creatures" , "Santa's sleigh puller" , "Domesticated animals" , "Antlered animal" .)