etymology - Similar 音・訓読み: Coincidence, or something more?

Occasionally, some 漢字 will have quite similar 音読み and 訓読み; such as, for instance 困; with 困{コン} and 困{こま}る; or 灰 with 灰{はい} and 灰{カイ}.

Are these cases exclusively coincidence, or are there situations in which Chinese words have been fully internalised as Japanese words rather than Sino-Japanese loanwords? (Or even; Japanese words exported to Chinese as fully internalised loanwords?)

Edit:

Thanks to naruto's comment; I remembered the one I really wanted to ask about (it was nagging at me as I wrote the question): 死, with 死{し}ぬ and 死{シ}.

Answer

Looking at the etymologies independently:

灰

This term appears in the Man'yōshū poetry compilation completed circa 759, so this is an older word. The older kana spelling of this was はひ, indicating an Old Japanese reading of /papi/.

Meanwhile, the on'yomi has an older kana spelling of くわい (probably realized as /kwai/), closer to the Middle Chinese it came from, which has a reconstructed reading of /xwoj/ -- think of the /x/ as a hard H like in Scottish loch, and think of the /j/ on the end like a Y.

The older kun'yomi of /papi/ doesn't overlap with the older on'yomi of /kwai/ in any way that I can find, so I think the similarity between modern on'yomi kai and kun'yomi hai is purely a coincidence of historical phonetic development.

困

The komaru reading with a sense of "stuck, troubled" only seems to appear from the early Edo period, from what I can find. I suspect this is an extension from / alteration of ancient root verb 込{こ}む "to go into something; to put something into something else", much like the English "stuck" can be interpreted as the passive participle form of root verb "stick" (as in, "to be stuck [into a bad situation]").

The older root's sense of "into" doesn't overlap much with the 困 kanji's meaning of "besieged, troubled", so I suspect this similarity is another coincidence of historical semantic development.

Other cases

Although the above two are coincidental, there are cases where the kun'yomi developed from the on'yomi. One example is 文{ふみ}, where the kun'yomi of fumi apparently came from an older on'yomi of pun. More at the Wiktionary entry.

In general, sound similarities are probably accidental. To find out for sure, you have to dig into the historical development of each term.