Irritation of church met with a smile (7)
I believe the answer is:
chagrin
'irritation' is the definition.
'church met with a smile' is the wordplay.
'church' becomes 'ch' (common abbreviation**).
'met' is a charade indicator (letters next to each other).
'with' means one lot of letters go next to another.
'smile' becomes 'grin' (grinning is a kind of smiling).
'ch'+'a'+'grin'='CHAGRIN'
'of' is the link.
(Other definitions for chagrin that I've seen before include "Arching with annoyed disappointment" , "Annoy" , "Emotion" , "irritation?" , "Disappointment, exasperation" .)