前文She considered herself lucky that she had insurance when she was diagnosed with breast cancer.回覆that子句作為adj及adv子句修飾lucky各一？可否再研判分析。?

這裡的that-clause是名詞子句, 作用是形容詞"lucky"的補語
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可以作為形容詞補語的, 一般有以下三種詞類 :
1. 介詞片語
e.g. We’ve been very lucky "with the weather". (朗文)
2. 不定詞片語
e.g. The children were lucky "to survive the fire" which destroyed their home. (朗文)
3. 名詞子句
e.g. She counted herself lucky "that she still had a job". (牛津)

She considered herself lucky----main clause
(that) she had insurance----noun clause
when she was diagnosed-----adv cl of manner
with breast cancer-----phrase.
Referring to your book by  TW Chen's the heart of grammar:-
Treating as noun clause:-eg:-
that she had the insurance with diagnosis.
Treating as adj. cl.at end:-eg:-
who had insured with breast cancer that was diagnosed--------ad cl at end meaning of reason, of way, of method,.
Treating as adv.cl:-eg:-
that she had insuranced when diagnosed----adv cl of time, of manner, of degree, of relation.