Is it permissible to supplicate in this manner: "O Allah, if the cough in my chest is good for me, then let it remain, and if it is evil for me, then cure me from it. My Lord, to You alone I have turned, and not to anyone other than You, and upon You I have relied (and I mention my supplication), My Lord, I am knocking at Your door and hoping for Your response"?

The aforementioned supplication is Islamically permissible because it is a submission to Allah's choice of what is good for His servant. Supplicating for well-being does not contradict patience; a Muslim can combine both. The Prophet, peace and blessings be upon him, used to guide the sick to supplicate. And abandoning treatment while enduring with patience, seeking reward, is better for whoever is able to do so, as happened with the woman who used to have epileptic fits. The Prophet, peace and blessings be upon him, encouraged asking Allah for pardon, well-being, and sustained health in this world and the Hereafter.