Little pest ran for cover after father turned up (5)
I believe the answer is:
aphid
'little pest' is the definition.
(insect considered a pest)
'ran for cover after father turned up' is the wordplay.
'ran for cover' becomes 'hid' (to run for cover is to run and hide).
'after' says to put letters next to each other.
'father' becomes 'pa' (pa can mean dad or father).
'turned up' says the letters should be written backwards.
'pa' reversed gives 'ap'.
'hid' after 'ap' is 'APHID'.
(Other definitions for aphid that I've seen before include "insect pest" , "one living on juices" , "Small insect like greenfly" , "Blackfly or greenfly" , "Garden bug" .)