A wood on the borders of a local river (7)
I believe the answer is:
spinney
'a wood' is the definition.
(I have seen 'little wood' mean 'spinney' so perhaps 'wood' could also mean 'spinney')
'on the borders of a local river' is the wordplay.
'on' is an insertion indicator (as in clothing 'on' a person).
'the borders of' means to remove the middle letters.
'a local' becomes 'native' (relating to a particular area).
'river' becomes 'spiny' (I can't justify this - if you can you should give a lot more credence to this answer).
'native' with its centre taken out is 'ne'.
'ne' going within 'spiny' is 'SPINNEY'.
Can you help me to learn more?
(Other definitions for spinney that I've seen before include "Small clump of trees" , "little wood" , "Small area of trees" , "Copse of trees" , "Cops, we hear" .)