Sudden chill: mild illnesses need sleep (4,4)
I believe the answer is:
cold snap
'sudden chill' is the definition.
(sudden drop in temperature)
'mild illnesses need sleep' is the wordplay.
'mild illnesses' becomes 'colds' (a cold is typically a minor illness).
'need' says to put letters next to each other.
'sleep' becomes 'nap' (synonyms).
'colds'+'nap'='COLD SNAP'
(Other definitions for cold snap that I've seen before include "Period of icy weather" , "Sudden short spell of bitter weather" , "Frosty spell" , "when snow may fall?" , "sudden brief spell of icy weather" .)