Not giving up with prettiness destroyed (10)
I believe the answer is:
persistent
'not giving up' is the definition.
(I've seen this before)
'prettiness destroyed' is the wordplay.
'destroyed' indicates an anagram.
'prettiness' anagrammed gives 'PERSISTENT'.
'with' is the link.
(Other definitions for persistent that I've seen before include "Relentless" , "Lasting, prolonged" , "Determined" , "Pressing" , "Continual" .)