Is it permissible to supplicate for marriage to a specific person, knowing that there is no legitimate relationship, and is this considered waiting for something that is not destined?

If the aforementioned young man is religious and of good character, it is permissible for the woman to supplicate to Allah to make him her husband. There is no harm in her informing him of her desire for him, though it might be preferable for this to be done through one of his female relatives. If the marriage is facilitated, then praise be to Allah. Otherwise, she should turn away from the idea, entrust her affair to Allah, ask Him for a righteous husband, and remember the words of Allah the Almighty: "And perhaps you hate a thing and it is good for you; and perhaps you love a thing and it is bad for you. And Allah knows while you know not." (Al-Baqarah: 216).