Most of each part of the golf-course staying fresh (9)
I believe the answer is:
evergreen
'staying fresh' is the definition.
'evergreen' can be an answer for 'staying' (thesaurus). I also know that 'fresh' relates to this answer.
'most of each part of the golf-course' is the wordplay.
'most of' means to remove the last letter.
'each' becomes 'every' (synonyms).
'part of the golf course' becomes 'green' (golf courses have greens).
'every' with its final letter taken away is 'ever'.
'ever'+'green'='EVERGREEN'
(Other definitions for evergreen that I've seen before include "Laurel, for example" , "kind of shrub" , "colourful" , "Source of wood" , "Durable" .)