Pursued, being grimly persistent (6)
I believe the answer is:
dogged
I believe this is a double definition.
'pursued' is the first definition.
(dogging is a kind of pursuing)
'grimly persistent' is the second definition.
(I've seen this before)
'being' is the link.
(Other definitions for dogged that I've seen before include "Relentless in a canine way" , "Worried" , "Follow closely - persistent" , "Tenacious, persistent" , "Resolute, tenacious" .)