Hi, Others may disagree, but I don't think desperation necessarily implies hopelessness. I think of it as trying every possibility in a sort of reckless way, caution to the wind - as opposed to calmly planning a strategy in an organized way, which will help you to achieve your goal. The "longing" implies deep emotion.
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desperate reflects a sense of hope; without which despair would be impossible; it is a feeling of hopelessness or being deprived of hope, from which .... without hope.... one would be utterly lost and absolutely overtaken. Even the violent and dangerous desire to fight which characterizes despair, is a desire and fight - not merely for hope - for something hope for.