You asked “is there a right or respectful way of praying in Buddhism?” I’d say that @OyaMist and @Brian_Dias_Flores were kind of on the right track, but I'd like to explore this deeper. Specifically, I’m wondering what it is you mean by the word “pray”? The Oxford English dictionary (and I'd guess most of us also) has 2 main definitions.
If you mean it in the sense of “to address God or a god with adoration, confession, supplication, or thanksgiving”, I’d say that it is not clear that the Buddha taught that there was a supreme being or multiple such (god(s)/God). In fact, in the discourse with Anathapindika, he seems to indicate that such does not exist. So I’d say in that sense a Buddhist does not pray. It would make no sense.
If you mean in the sense of “wish or hope strongly for a particular outcome or situation”, I’d say that this might be a way of describing meditation; although I’m not sure I’d use the words “wish” or “hope” here. Maybe more “seek for” would be better used for what you do when you meditate.
The net of it is that I’d suggest that a Buddhist does not “pray” at all; as “meditation” is something completely different from “prayer” in the most common meanings.
With regard to the mantras used in the meditative process, I was taught that these are not designed to be communications to any outward being or entity. Rather, they were designed to be repetitiously repeated to, by sound and thought, be aimed inward in order to help align the body and mind and intensify the meditative process. Nonsense syllables can be used as well as repeated “words” so I’m not sure you could consider them “prayers” in any meaningful sense.
I am admittedly, though, a lay person who is struggling with understanding the teachings of the Buddha myself and would appreciate correction by any who have better knowledge in these areas.