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Thank you to all contributors. If the word "entryway" appears in the OED, then obviously I have to accept it exists. I totally accept "approach" indicates way, which means the definitionhas to be "Perhaps US door". dannyk13 refers to "noun NORTH AMERICAN, a way in to somewhere or something; an entrance" - which would seem to settle the matter, though I would be interested to know what the source for that is.
My only version of the OED is a very heavy 2 volume version (in tiny print with a magnifying glass!) entitled THE COMPACT EDITION OF THE OED, which was given to me as a wedding present in 1984. I did check it but it had nothing.
My primary reference has always been Chambers, (probably because of getting into The Listener many decades ago), and I was surprised that they didn't have it, and that Google produced nothing. I'm not tech savvy enough to know how to source all the other dictionaries that are out there online, but if anyone is prepared to give me a few pointers, I would very much appreciate it.
Thanks again to all. I give Best Answer to dannyk13, both for speed and clarity as regards which precise bit of the clue was the definition.