Small number perished outside, ignored (10)
I believe the answer is:
discounted
'ignored' is the definition.
('discount' can be a synonym of 'ignore')
'small number perished outside' is the wordplay.
'small' becomes 's' (abbreviation - e.g. clothes size).
'number' becomes 'count' (count is a kind of number**).
'perished' becomes 'died' ('die' can be a synonym of 'perish').
'outside' means one lot of letters goes inside another.
's'+'count'='scount'
'scount' inserted inside 'died' is 'DISCOUNTED'.
(Other definitions for discounted that I've seen before include "Ignored as insignificant" , "not considered" , "Like relatively cheaper goods" .)