Is it true that this supplication was authentically narrated from the Prophet, peace and blessings be upon him: "O Allah, look upon us with the eye of Your mercy," and is there any issue with the word "eye"?

There is no authentic Hadith with this exact wording narrated from the Prophet, peace and blessings be upon him. However, using the word "ayn" (eye) in the singular form is permissible because a singular noun in an annex construction does not preclude plurality, as in His (Allah's) saying, "that you might be brought up under My eye" [Quran 20:39], and as in His saying, "And if you should count the favors of Allah, you could not enumerate them" [Quran 14:34].