Behaved abysmally chasing them every day at first (5)
I believe the answer is:
acted
'behaved' is the definition.
(acting is a kind of behaving)
'abysmally chasing them every day at first' is the wordplay.
'at first' indicates taking the first letters.
The first letters of 'abysmally chasing them every day' is 'ACTED'.
(Other definitions for acted that I've seen before include "Performed (in a play)" , "Appeared in play" , "Cadet (anag)" , "Worked" , "exerted influence" .)