Really important to get on someone's nerves on air (5)
I believe the answer is:
great
'really important' is the definition.
(great can mean important or significant)
'to get on someone's nerves on air' is the wordplay.
'to get on someone's nerves' becomes 'grate' (grate can mean to annoy someone).
'on air' indicates a 'sounds like' (homophone) clue (how the word sounds on the radio, on air).
'grate' is a homophone of 'GREAT'.
(Other definitions for great that I've seen before include "Peter the -, Russ. tsar" , "Large, outstanding" , "and 12 Down: Powerful dog" , "Large, important" , "Wonderful" .)