precious thing - ?

Думаю (of) sentimental value - это как раз "дорого сердцу"

"The 21-year-old actor recently finished shooting the final scenes of “Harry Potter and the Deathly Hallows”, the seventh and last of the films based on the eponymous boy wizard whom he has played for 10 years, and said his prop glasses were the only thing of sentimental value he wanted to keep."

Спасибо, буду знать.)

https://dictionary.cambridge.org/dictionary/english/coveted - strongly desired by many
https://www.collinsdictionary.com/dictionary/english/coveted - something that very many people would like to have.

С какого перепугу это можеть означать "дорогое сердцу", знает только Валерий.

Valery, COVETED has a negative connotation in that you want what others have and even begrudge them for it.