Refuse request ultimately not considered (5)
I believe the answer is:
trash
'refuse' is the definition.
(both can mean rubbish)
'request ultimately not considered' is the wordplay.
'ultimately' indicates one should take the final letters.
'not considered' becomes 'rash' (thoughtless or not considered properly).
The final letter of 'request' is 't'.
't'+'rash'='TRASH'
(Other definitions for trash that I've seen before include "Rubbish, waste" , "pulp" , "Take the tops from < rhubarb" , "Condemn" , "Worthless people; destroy (US slang)" .)