Publicise footwear aid for a diver? (3,4)
I believe the answer is:
air pump
'aid for a diver?' is the definition.
(an air pump might help a diver to breathe)
'publicise footwear' is the wordplay.
'publicise' becomes 'air' (airing is a kind of publicising).
'footwear' becomes 'pump' (pump is a kind of footwear).
'air'+'pump'='AIR-PUMP'
(Other definitions for air pump that I've seen before include "What can be used to create a vacuum" , "Cause of inflation" , "It inflates" .)