Resolute quality of head chasing dates around hurriedly (13)
I believe the answer is:
steadfastness
'resolute quality' is the definition.
'head chasing dates around hurriedly' is the wordplay.
'head' becomes 'ness' (ness is a headland**).
'chasing' means one lot of letters go next to another.
'around' is an anagram indicator.
'hurriedly' becomes 'fast'.
'dates' is an anagram of 'stead'.
'stead'+'fast'='steadfast'
'ness' after 'steadfast' is 'STEADFASTNESS'.
'of' is the link.
(Other definitions for steadfastness that I've seen before include "Resolution" , "Firm unwavering quality" , "Resolute attitude" , "Loyalty" .)