Cause not to hear new promise being brought up (5)
I believe the answer is:
drown
'cause not to hear' is the definition.
(as in drowning out a sound)
'new promise being brought up' is the wordplay.
'new' becomes 'n' (common abbreviation eg NT for New Testament).
'promise' becomes 'word' (as in 'I give you my word').
'being brought up' shows that the letters should be reversed in order (in a down clue, letters go up).
'n'+'word'='nword'
'nword' reversed gives 'DROWN'.
(Other definitions for drown that I've seen before include "Immerse (fatally?)" , "Completely conceal" , "Kill by submersion" , "Die by immersion in liquid" , "Go under" .)