"strong longing" is a correct and usable phrase in written English.
You can use it to describe a feeling of intense desire for something, either physical or abstract. For example, "I have a strong longing to travel the world."
They are people who have vague but strong longings for something like secure employment, and no idea where to look for it; who on the whole want to belong, and live in a climate where they are taken seriously as workers, as citizens – and as needy individuals; and who have got used to being pushed to the margins and told that they are dispensable.
However, consider the idea that the impulse of that strong emotion, longing, or pain could be the unexpressed desire of your soul essence (that place within us where Infinite presence has personalized itself) seeking our attention.
It's an audience group that's perhaps less sophisticated but has strong materialistic longings – and Tiny Times caters very skillfully to it.
Maybe the strong emotions, longings, and even the pain, arise in those moments when we need to be reminded of the one truth that we tend to most often forget; perhaps it is the beloved wanting to tell us we are never alone, irrespective of where we are or what the circumstances may be.
Be warned: you are about to experience strong feelings of longing for Barry Davies.
The urge to mock our actual presidents is as strong as our longing, in the abstract at least, to revere the president.
Alice has grown older and more knowing than her original model, and turned into the receptacle of erotic dreams, a femme enfant with whom women artists strongly identify: the knowledge you are Alice as strong as the longing for her.
Have you ever had a strong emotion or longing -- or even a twinge of pain -- arise in the field of your awareness, and then quickly brushed it off, continuing on with whatever you were doing rather than sitting with it and allowing it to speak to you?
Mr. Almodóvar's characters tend to be stricken with their own versions of the malady — subject to strong and confused longings, surprised by pain in their pursuits of pleasure.
