Mine closes branch that’s a bad place to be (6)
I believe the answer is:
armpit
'a bad place to be' is the definition.
(armpit can mean an unpleasant place)
'mine closes branch' is the wordplay.
'mine' becomes 'pit' (down the pit).
'closes' means one lot of letters go next to another.
'branch' becomes 'arm' (synonyms).
'pit' after 'arm' is 'ARMPIT'.
'that's' is the link.
(Other definitions for armpit that I've seen before include "A sort of hollow" , "depression which is personal" , "Hole for a limb?" , "Where to put a crutch" , "Depression under an upper limb" .)