Which is the correct usage of the following? YESHUA HaMASHIACH or Jesus Christ?
Which would be correct depends on your audience and the language in which you are speaking. A Messianic Jewish audience would understand “Yeshuah haMoshiach” (or haMashiach) with no problem. Most non-Jewish Christian audiences would be confused by the Hebrew name, unless it were a Hebrew-speaking audience, but would understand “Jesus Christ” or “Jesus the Christ.” If speaking primarily in English, “Jesus Christ” would be acceptable. If speaking in Hebrew, the Hebrew name would be more appropriate. If speaking Spanish, “Jesucristo” would be more appropriate than either of your choices. To an audience that commonly uses “Jesus Christ” as a swear word, then “Jesus the Christ” or “Jesus the Messiah” might avoid confusion.
If speaking to a Jewish audience (other than Messianic Jews), then either term would be offensive. More appropriate would be Yeshua ben Miryam (Jesus the son of Mary), or any other proper appellation that does not refer to him as Messiah/Christ. One would get further in teaching the Jewish people about Jesus by not using a phrase that would automatically put up barriers. “Christ” and “Christian,” especially, have such negative connotations to the Jewish mind that they should never be used, possibly even if the person has been immersed into Jesus as the Messiah.