… new animal sanctuary — first to accept incarceration, essentially will make most irritable (8)

'most irritable' is the definition.
(narky can mean bad-tempered)

'new animal sanctuary first to accept incarceration essentially' is the wordplay.
'new' becomes 'n' (common abbreviation eg NT for New Testament).
'animal sanctuary' becomes 'ark' (as in Noah's Ark).
'first to' becomes 'ist' (I've seen this in other clues).
'accept' is an insertion indicator.
'essentially' means to look at the middle letters.
The middle of 'incarceration' is 'e'.
'ist' enclosing 'e' is 'iest'.
'n'+'ark'+'iest'='NARKIEST'

'will make' is the link.