I want to know whether we can supplicate for or on behalf of non-Muslims like Christians and the likes, and if we can, apart from supplicating for their being guided to Islam, can we supplicate for their sick ones or for, their poor ones to be endowed with Rizq and the likes.

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

Yes, it is permissible to supplicate to Allaah to guide the non-Muslims. The Prophet said: "O Allaah! Guide the tribe of Daws." [Al-Bukhari and Muslim] Furthermore, Imaam At-Tirmithi reported that the Jews used to sneeze in front of the Prophet expecting him to tell them "May Allaah have mercy on you", but the Prophet used to tell them: "May Allaah guide you and rectify your affairs."

However, it is not permissible to seek mercy and forgiveness for them, as Allaah says (which means): {It is not (proper) for the Prophet [] and those who believe to ask Allaah's Forgiveness for the Mushrikeen (polytheists, idolaters, pagans, disbelievers in the Oneness of Allaah) even though they be of kin.}[Quran 9:113]. Also, the Prophet refused to seek forgiveness for them. As regards supplicating for them for worldly benefits, the scholars differed in opinion about it. Nonetheless, it is reported in one narration that it is permissible to make a supplication for them as regards worldly matters. The Prophet said: "If you supplicate for a Jew or a Christian say: 'May Allaah increase your progeny and your wealth.'" [Ad-Daylami]

Allaah knows best.