At last, it's do or die – failing another way out (4,4)
I believe the answer is:
side door
'another way out' is the definition.
Although both the answer and definition are singular nouns, I can't see how one could define the other.
'at last it's do or die failing' is the wordplay.
'at last' suggests the final letters.
'failing' indicates anagramming the letters (a failed spelling of the word).
The last letter of 'its' is 's'.
'do'+'or'+'die'='doordie'
'doordie' with letters rearranged gives 'idedoor'.
's'+'idedoor'='SIDE-DOOR'
Can you help me to learn more?
(Other definitions for side door that I've seen before include "less obvious entrance" , "Unexpected way to get in" , "Another way in" , "Entrance round the corner of the house" , "Minor entrance" .)