Hi Samantha, This is used to describe that time is extremely valuable and is often used to indicate if you are short of time for some particular task. The word precious here indicates "not very much" and is only used in the positive. Note that we do not use an article. "We have precious time left"

간절하다 means "ardent, earnest, desperate" 소중하다 means "precious, valuable, dear" 간절하고 소중한 시간 the earnest and precious time

Yes it means literally '소중한 시간' :) but for more exact meanings you need to comprehend it with a sentence.