One cat ran off straight away (2,4)
I believe the answer is:
at once
'straight away' is the definition.
'at once' can be an answer for 'away' (thesaurus). I am unsure of the 'straight' bit.
'one cat ran' is the wordplay.
'ran' indicates an anagram (letters have run around).
'one'+'cat'='onecat'
'onecat' with letters rearranged gives 'AT ONCE'.
'off' is the link.
Can you help me to learn more?
(Other definitions for at once that I've seen before include "without pause" , "This instant - no delay (2,4)" , "Right now, this instant" , "This instant, without delay" , "without hesitation" .)