Poor old American introduced to lady without notice (5)
I believe the answer is:
lousy
'poor' is the definition.
(lousy can mean of poor quality)
'old american introduced to lady without notice' is the wordplay.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'american' becomes 'us' (abbreviation for United States).
'introduced to' is an insertion indicator.
'without' indicates named letters should be taken away.
'notice' becomes 'ad' (short for advert).
'lady' with 'ad' taken away is 'ly'.
'o'+'us'='ous'
'ous' put within 'ly' is 'LOUSY'.
(Other definitions for lousy that I've seen before include "Infested by bloodsuckers" , "Infested: mean" , "Pants" , "crawling with insects" , "Second-rate" .)