Most sickly-looking Gunners seen around a training facility (9)
I believe the answer is:
palaestra
'training facility' is the definition.
(I know that palaestra is a type of training facility)
'most sickly-looking gunners seen around a' is the wordplay.
'most sickly looking' becomes 'palest'.
'gunners' becomes 'RA' (Royal Artillery).
'seen around' is an insertion indicator.
'palest'+'ra'='palestra'
'palestra' enclosing 'a' is 'PALAESTRA'.
(Other definitions for palaestra that I've seen before include "In old athletics arena" , "gymnasium" , "Greek training school" .)