Clue: Breathing apparatus worn by Greek character in British 'ovel (10)
Answer: BRONCHIOLE
Explanation: Definition: Breathing apparatus
**ON** ("worn by") + **CHI** ("Greek character" (a letter in the Greek alphabet)) in **Br**.(itish) (h)**OLE** ('ovel), so **Br.(ON-CHI)OLE** 
*A bronchiole is a tiny tube inside your lungs.