Correct spelling for the English word "bradway" is [bɹˈadwe͡ɪ], [bɹˈadweɪ], [b_ɹ_ˈa_d_w_eɪ] (IPA phonetic alphabet).
The word "bradway" does not have historical etymology as it appears to be a variation or misspelling of the term "Broadway". However, "Broadway" itself has an interesting etymology. It originates from the Old English words "broad" (meaning wide) and "weg" (meaning way or road). In the early 17th century, "Broadway" referred to a wide street in the city of Westminster, London. The name transferred to New York City's famous theater district in the late 18th century, which was named after the road that ran through it. Over time, "Broadway" became synonymous with the American theater industry.