Serious offence outside public house by puzzler (6)
I believe the answer is:
sphinx
'puzzler' is the definition.
(I've seen this before)
'serious offence outside public house by' is the wordplay.
'serious offence' becomes 'sin' (I've seen this before).
'outside' is an insertion indicator.
'public house' becomes 'ph'.
'by' becomes 'x' (multiplication sign).
'sin' going around 'ph' is 'sphin'.
'sphin'+'x'='SPHINX'
(Other definitions for sphinx that I've seen before include "Stone figure by the pyramids at Giza" , "ancient monument" , "Egyptian figure at Giza" , "puzzler" , "mythical monster" .)