Means to cover boobs – a new cycling trimmer (8)

I haven't cracked this one yet — but the brilliant team at Fifteensquared have:

A N (a new) TIPPEX (means to cover boobs) with the letters cycling; I knew the word (I’d spell it Xanthippe) – a nagging woman, from Socrates’ wife, but I was stuck for the definition, until I found ‘a scold’ for ‘trimmer’ in Chambers