Excuse having run over priest the night before (7)
I believe the answer is:
relieve
'excuse' is the definition.
(I know that excuse can be written as relieve)
'run over priest the night before' is the wordplay.
'run' becomes 'r' (cricket abbreviation).
'over' means one lot of letters go next to another (in a down clue, some letters go over others).
'priest' becomes 'eli' (biblical figure).
'the night before' becomes 'eve' (eve can mean the night before something).
'r'+'eli'+'eve'='RELIEVE'
'having' acts as a link.
(Other definitions for relieve that I've seen before include "Alleviate, assuage" , "Assuage, ease" , "raise a siege" , "Soothe" , "help" .)