1. Jill was afraid she had forgotten her key at home, but she found it in her handbag. 2. Dad wasn’t at home when I came back. He had gone out twenty minutes before. 3. I wasn’t hungry because I had just had breakfast. 4. Peter saw an urgent message on his table. Somebody had left it the day before. 5. I apologized I had not phoned her. 6. He told me that he had come back a fortnight before. 7. I knew him at once though I had met him many years before. 8. We spent the night in Klin, a town we had often heard of but had never seen. 9. They couldn’t believe he had given up his job in the bank. He had made a good living there. 10. Mr. Jackson said that he had already bought everything for lunch. 11. Alice asked her brother where he had arranged to meet his friends. 12. We had no car at that time because we had sold our old one. 13. They had finished painting the ceiling by two o’clock.

Ниже — проверка каждой фразы с правками (если нужны) и коротким объяснением, зачем здесь Past Perfect и где лучше обойтись другими формами.

1. ✔️ Jill was afraid (that) she had forgotten her key at home, but she found it in her handbag.
   Past Perfect (had forgotten) указывает на более раннее действие относительно прошедшего «was afraid»/«found». Всё ок.

2. ✔️ Dad wasn’t at home when I came back. He had gone out twenty minutes before.
   «Had gone out» произошло раньше «came back». Нормально.

3. ✔️ I wasn’t hungry because I had just had breakfast.
   Двойное «had» корректно: Past Perfect от «have». Завтрак был раньше состояния «wasn’t hungry».

4. ? (стилистика) Peter saw an urgent message on his desk. Somebody had left it the day before.
   Грамматика верная; «desk» естественнее, чем «table». Past Perfect показывает, что записку оставили до того, как Питер её увидел.

5. ? Лучше так: I apologized to her for not calling (her). / I apologized for not having phoned her.
   Конструкция apologized that… неестественна. Здесь не про последовательность прошедших действий, а про факт извинения → обычный Past Simple; для «не позвонил раньше» — герундий («for not calling / for not having phoned»).

6. ✔️ He told me that he had come back a fortnight before.
   Сообщение в прошедшем, а «вернулся» — ещё раньше, потому Past Perfect уместен.

7. ? Лучше по смыслу: I recognized him at once, though I had met him many years before.
   «Knew him» — «знал его» (знаком был), а нужна «узнал» → recognized. Past Perfect для встречи много лет назад — ок.

8. ✔️ We spent the night in Klin, a town we had often heard of but had never seen.
   Действия «слышали/не видели» предшествуют «провели ночь», Past Perfect уместен.

9. ✔️ They couldn’t believe he had given up his job in the bank. He had made a good living there.
   Оба действия (уволился; хорошо зарабатывал) ранние по отношению к «couldn’t believe». Нормально.

10. ✔️ Mr. Jackson said that he had already bought everything for lunch.
    Сообщение в прошедшем, а покупка — раньше → Past Perfect.

11. ✔️ Alice asked her brother where he had arranged to meet his friends.
    Спросила (прошедшее), договорился — раньше → Past Perfect.

12. ✔️ We had no car at that time because we had sold our old one.
    Причина («продали раньше») оформлена Past Perfect — логично.

13. ⚖️ Зависит от контекста:
    — Если есть последующая точка отсчёта в прошлом: They had finished painting the ceiling by two o’clock when we arrived. (Past Perfect оправдан, «к моменту нашего прихода» уже закончили.)
    — Если фраза стоит сама по себе: естественнее They finished painting the ceiling by two o’clock. (обычный Past Simple + «by»).

Итог: почти все предложения корректны с Past Perfect; ключевые правки — №5 (управление после apologize), №7 (semantics: recognized), стилистика №4 (desk), и уточнение по №13 (нужна «вторая» прошедшая точка, иначе Past Simple).