Person looking round briefly to annul or lower offer (8)

'lower offer' is the definition.
(informal term meaning to lower an offer of sale)

'person looking round briefly to annul' is the wordplay.
'person looking' becomes 'gazer'.
'round' means one lot of letters goes inside another.
'briefly' means to remove the last letter.
'to annul' becomes 'undo' (I've seen this before).
'undo' with its final letter taken away is 'und'.
'gazer' placed around 'und' is 'GAZUNDER'.

'or' is the link.