His fate disheartened men working in part of pit (9)
I believe the answer is:
mineshaft
'part of pit' is the definition.
('mineshaft' is part of 'pit')
'his fate disheartened men working' is the wordplay.
'disheartened' suggests removing the centre (the middle or heart is removed).
'working' is an anagram indicator.
'men' with its centre taken out is 'mn'.
'his'+'fate'+'mn'='hisfatemn'
'hisfatemn' is an anagram of 'MINESHAFT'.
'in' is the link.
(Other definitions for mineshaft that I've seen before include "one's got bored" , "Vertical hole" , "not a feature of 4, 21 [OPENCAST MINING]" .)