Cutting surreptitious chant on the radio (7)

'cutting' is the definition.
(I know that cutting can be written as slicing)

'chant on the radio' is the wordplay.
'chant' becomes 'sing' (chant can mean to sing).
'on' is an insertion indicator (as in clothing 'on' a person).
'the radio' becomes 'lic' (I can't explain this - if you can you should give a lot more credence to this answer).
'sing' placed around 'lic' is 'SLICING'.

'surreptitious' acts as a link.
I am not very happy about this link. It may be part of another bit of the clue.