Animal's down in the dumps right away (5)
I believe the answer is:
moose
'animal's' is the definition.
(thesaurus)
'down in the dumps right away' is the wordplay.
'down in the dumps' becomes 'morose' (I've seen this before).
'right' becomes 'r' (common abbreviation).
'away' indicates named letters should be taken away.
'morose' with 'r' taken away is 'MOOSE'.
(Other definitions for moose that I've seen before include "Large northern deer" , "Animal" , "Large N American deer like European elk" , "Large US deer" , "North American deer" .)