Does it matter to you that theourgia is a late ancient Greek word, from the Roman imperial period, fostered by Neoplatonists? That is, it’s not a Homeric term or concept. I understand that “realism” is not the goal (ha!) but capturing the spirit of Epic seemed to be one of the goals.
Prayer, in Homeric Greek, was λιτή or εὐχή (the latter being more along the lines of what you have). Again, I don’t assume you are trying to make it “authentic,” and there is no need to do so, but you did express the wish to capture a certain flavor linked to a historical period.
“Magic” didn’t exist yet by that term. Ancient Greeks described the rituals of the magi (Zoroastrian priests from the Persian Empire, a post-Homeric empire) as “magic” (related to those priests). From there, any hocus pocus could be described as “magic.”