Whenever my daughter gets sick, her throat starts to bother her before anything else and her cough lingers.
My daughter's throat is a little delicate, so her cough always stays around for a while when she gets a cold.
ーWhenever my daughter gets sick, her throat starts to bother her before anything else and her cough lingers.
「娘は病気になるとまず喉にくるみたいで、咳が長引きます」
her throat starts to bother her before anything else「何よりも先に喉が悩まし始める」＝「まず喉にくる」
to linger で「なかなか消えない・去らない」
ーMy daughter's throat is a little delicate, so her cough always stays around for a while when she gets a cold.
「娘の喉はちょっと弱くて、風邪をひくといつも咳が長引く」
delicate を使って喉が弱いことを表現できます。
her cough always stays around for a while「彼女の席はいつもしばらくとどまる・居座る」＝「咳が長引く」
ご参考まで！
Since a long time her throat has been a bit weak so that's why her cough always lingers.
ご質問ありがとうございます。
少し長い言い方ですが、「元々喉が弱くいつも咳が長引きます」と言いたいなら、英語で「Since a long time her throat has been a bit weak so that's why her cough always lingers.」と言えます。
これで娘の咳について、先生にはっきり伝えられます。
ご参考になれば幸いです。
カナダ 
アメリカ合衆国 
