Cover over public pool (4)
I believe the answer is:
lido
'public pool' is the definition.
(I've seen this before)
'cover over' is the wordplay.
'cover' becomes 'lid' (lid is a kind of cover**).
'over' becomes 'o' (cricket abbreviation).
'lid'+'o'='LIDO'
(Other definitions for lido that I've seen before include "Aquatic leisure centre" , "Public open swimming pool" , "Swim here" , "Italian bathing place" , "Venetian resort" .)