What is the ruling on saying "God the Generous" (جو الكريم) or "Upon the Generous" (على الكريم) when inquiring about one's health, and is it permissible to supplicate with lawful wordings that were not narrated from the Messenger of Allah, peace and blessings be upon him?

It is best to supplicate with the ma'thur (transmitted supplications), but it is permissible to supplicate with other words so long as they do not involve aggression, a supplication for sin, or the severing of kinship ties. Evidence for this is the Prophet's (peace be upon him) approval of some Companions' supplications that were not ma'thur, and the general meaning of the hadith in Muslim: "The servant will be answered as long as he does not supplicate for sin or the severing of kinship ties." As for the phrase "ʿala al-karīm" (dependent on the Generous One) or "jawwāl karīm" (the Generous One's atmosphere) in response to a question about one's health, there is no harm in it if the intention is to rely on Allah or that one is under His care.