Professor has an employee starting who is quite irreligious (7)
I believe the answer is:
profane
'irreligious' is the definition.
(I've seen this in another clue)
'professor has an employee starting' is the wordplay.
'professor has' becomes 'prof' (I can't justify this - if you can you should believe this answer much more).
'starting' indicates taking the first letters.
The first letter of 'employee' is 'e'.
'prof'+'an'+'e'='PROFANE'
'who is quite' acts as a link.
This may not be right. Some or all of it may be part of another bit of the clue.
Can you help me to learn more?
(Other definitions for profane that I've seen before include "Impious" , "(Of language) blasphemous" , "Irreverent - pagan" , "Secular; obscene" , "Secular, not religious" .)