Ungodly teacher an irrelevance, ultimately (7)
I believe the answer is:
profane
'ungodly' is the definition.
(thesaurus)
'teacher an irrelevance ultimately' is the wordplay.
'teacher' becomes 'prof' (I've seen this in other clues**).
'ultimately' indicates one should take the final letters.
The final letter of 'irrelevance' is 'e'.
'prof'+'an'+'e'='PROFANE'
(Other definitions for profane that I've seen before include "vulgar" , "Secular, not religious" , "Contemptuous of sacred things" , "Blasphemous, irreligious" , "Sacrilegious or blasphemous" .)