It may be close to the church, or a serious distance (9)
I believe the answer is:
graveyard
'it may be close to the church' is the definition.
The definition suggests a singular noun which matches the answer.
'a serious distance' is the wordplay.
'a serious' becomes 'grave' (both can mean solemn).
'distance' becomes 'yard' (I've seen this before).
'grave'+'yard'='GRAVEYARD'
'or' acts as a link.
Can you help me to learn more?
(Other definitions for graveyard that I've seen before include "'God's acre'" , "Cemetery" , "church ground" , "remains here" , "Final destination" .)