Does saying, "I wish, my Lord," or "If only, my Lord," imply that the supplication will not be answered, and is it preferable to say, "I implore You, my Lord," instead?

The word "أتمنى" (I wish/hope) in supplication is a linguistic matter, not a jurisprudential one. Religion does not prohibit supplicating with it. "أتمنى" is used in the Arabic language for matters difficult or impossible to occur, while "أرجو" (I hope/implore) is for hoping in what is possible. When supplicating, a servant should ask with certainty and good presumption of Allah, which is not compatible with expressions of wishing. Therefore, it is preferable to use "أرجو" or "أسأل الله" (I ask Allah). However, there is no harm for one who uses "أتمنى" due to their linguistic habit, for Allah accepts supplications from a sincere heart, whether the wording is eloquent or not.