I was going through the book The Languages of Japan and Korea, by Nicolas Tranter, and thought it would be a fun mental exercise to try and connect some of the Old Korean reconstructions that were provided and connect them to Japanese words. Of course, there were quite a few, especially when it came to the Koguryo dialect (so-called in the book). But the ones that stood out to me were the words for "stream", "star" and "old times" in Old Korean.
*nari > nay > nay 'stream'
*niäri > nyey > yey 'old times'
*piäli > pyel 'star'
In a previous discussion, I remarked that the sequences *l, *li and *ri in Korean seemingly become *shi or *ri in Japanese, and we often see what look like doublets in the language (e.g. bakkari vs bakkashi). Now, if we compare the same words above in Japanese, we get the following:
nagare "stream" (also: nagashi "pouring", na "river")
mukashi "old times"
*poshi > hoshi "star"
In the two first words, it's interesting that Japanese has three syllables where modern Korean now only has one. Also curious is the fact that the endings closely match up per the above-mentioned sound correspondence, and the first consonant is a nasal. If the words were related, then it becomes evident that Old Korean would have underwent a sound change that lenited and deleted the medial /k/ (*nakari > *nari; *nikäri > *niäri).
Now, if we assumed by parallelism that the word for "star" in Korean underwent the same change, we would obtain *pikäli > *piäli. Curiously, this seems very similar in shape to the Japanese word *pikari > hikari "light". We also know that there's variance between the syllables /hi/ and /ho/ in Japanese, namely in the word for "fire", so a pseudo Proto-Japanese word *pokari~*pokashi wouldn't be unthinkable. Perhaps here too the /ka/ disappeared in Japanese.
If all of these assumptions were correct, then the Japanese and Korean words would be cognates and the word for "star" in Japanese could be related to "light". Of course, as always, these are just my thoughts and may prove to be untrue. Remember also that borrowings are possible, so any word connection doesn't mean the two languages are related.