Part supporting heartless undertaking (8)
I believe the answer is:
fraction
'part' is the definition.
(fraction is a kind of part)
'supporting heartless undertaking' is the wordplay.
'supporting' becomes 'for'.
'heartless' means to remove the middle letters.
'undertaking' becomes 'action'.
'for' with its middle removed is 'fr'.
'fr'+'action'='FRACTION'
(Other definitions for fraction that I've seen before include "A division of a whole number" , "Small piece forming part of a whole" , "little bit" , "Small proportion" , "Fragment" .)