Is it sufficient for a Muslim to ask for good in general in his supplications, or should he specify his request, and is supplicating for healing from diseases for which there is currently no cure considered an transgression in supplication?

There is no blame upon a Muslim for seeking healing from illness; it is permissible and does not contradict general supplications. Many prophetic traditions indicate this, among them the supplication of the Prophet, peace be upon him, for Sa'd's healing.

Seeking healing from any illness, such as cancer or others, is not an transgression in supplication and is not impossible. Indeed, Allah Almighty has not sent down any disease without also sending down a cure for it, as stated in the hadith: "Treat yourselves, O servants of Allah, for Allah Almighty has not placed a disease without also placing a cure for it, except for old age."