When I travelled, I found I could translate the regions or cities into names in English,
Do the regions of Japan translate similar? Are there any references to their English names, for example "region of blossoms" or such?
By provinces I mean provinces like Hōki, Inaba, Harima, Bizen, Mimasaka, Bitchū, etc.?
Most Chinese and Japanese place names are made of kanji, which is perhaps the most famous sets of logographic characters currently in use. Therefore most Japanese place names can be translated literally, just as Shanghai (上海) translates to "upon-the-sea" (上 = up, 海 = sea). For example, this page has a list of the literal "translations" of the 47 Japanese prefectures.
Given that place names are usually written with kanji (Chinese logographic characters), it is hard to find a place name that doesn't have a meaning. The "meaning" of a place name is hard-coded into the way it is written.
Often the characters seem to represent the etymology of a place name (京都 "capital city", 大阪 "large hill", etc.).
However, sometimes the characters may have been selected for their phonetic value instead. (This is especially true for old names, from a time when kanji were often used for their phonetic value and less for their meaning.)
For example, in the case of the old province 播磨国 Harima you asked about, there was more than one way of writing the name: Wikipedia mentions also 針間国 and 幡麻国. Apparently, the origin of the name Harima is unknown. However, with each different way of writing the name, a different meaning is attached:
(I don't know how to accurately translate the names.)
Similarly, 美作国 Mimasaka no Kuni "Land of Artwork" (remember I'm just making this up) is written with characters that have little to do with the purported etymology of the place name (御坂の国 "Land of the Sacred Hills", 三坂の国 "Land of the Three Hills", 甘酒の国 "Land of Amazake").
In general place names may have adopted different characters for some such reasons. For translation, you have to choose a representation that you use to translate "the" meaning into English.
