Hoylake's third, middle of green — I fancy chance of eagle here (5)
I believe the answer is:
eyrie
'chance of eagle here' is the definition.
(an eyrie is the nest of a large bird)
'hoylake's third middle of green i fancy' is the wordplay.
'hoylake's third' becomes 'y' (3rd letter of 'hoylake').
'middle of' indicates the central letters.
'fancy' indicates anagramming the letters.
The central letters of 'green' are 'ree'.
'y'+'ree'+'i'='yreei'
'yreei' is an anagram of 'EYRIE'.
(Other definitions for eyrie that I've seen before include "Eagle's lair" , "home of 12 [RAPTOR]" , "Lofty home" , "Inaccessible place" , "retreat for hunter" .)