Degree of spirit regarding publicity shown in check for errors (9)
I believe the answer is:
Here is my best explanation:
'check for errors' is the definition.
'degree of spirit regarding publicity' is the wordplay.
'degree of spirit' becomes 'proof' (term for the strength of an alcoholic spirit).
'regarding' becomes 're'.
'publicity' becomes 'ad' (advert).
'shown in' is the link.
(Other definitions for proofread that I've seen before include "correct text", "Carefully check", "Prepare for publication" and "Checked text".)