Is it permissible for a man who has two wives, and is unable to provide for both of them, to keep the wife he prefers, or must he keep the first wife, especially if the first wife desires a Khul'?

If you are unable to provide for both of your wives, then you are permitted to separate from one of them by your choice, without preferring the first over the second, based on the hadith (definition) of Ghaylan ibn Salamah al-Thaqafi. Since the reason for separation is your inability to provide, you must divorce one of them and not seek khul' from her, as it is not permissible for the husband to take money from his wife in khul' if the reason for separation is from his side. It is also worth noting that the second marriage was hasty, and it is preferable to try to find work and strive to earn the maintenance to keep both wives together.