pronunciation - When Chinese personal names are written in Japanese in kanji is there always an obvious reading?

In their comment to an answer on the question "Are foreign personal names usually written in katakana rather than Romaji?", user sawa says:

... Chinese names should be written in kanji rather than katakana and read by the Japanese pronunciation. For example, 金大中 is キムデジュン, not きんだいちゅう, but 毛沢東 is もうたくとう, not マオジードン. ...

This got me wondering, since each Japanese character generally has at least two readings is there always one clear reading for pronouncing such names?

I expect of course that the "on" readings would be used, but often there is more than one on reading for the same character.

From the comments so far (no answers yet) this is indeed very interesting, and on Zhen Lin's prompting I would like to include Korean names as well despite their being discounted in the older linked question.

Answer

According to Wikipedia,

[...]

また、朝日新聞は中国人名のルビを中国語読みで表記している。（グループ会社のテレビ朝日は日本語読み）

[...]

日本漢字音による読みは原則として漢音を用いるが、金日成、済物浦、銭其琛をそれぞれ「きんにっせい」、「さいもっぽ」、「せんきしん」と読むように呉音や慣用音が用いられることも稀にある。また個別の慣用によって、北京をペキン、香港をホンコンと読んだり、台湾の高雄を「たかお」と訓読みしたりする場合がある。

Here is my translation, additions in square brackets:

[...]

On the other hand, the Asahi Shimbun prints Chinese names with ruby annotation giving the Chinese pronunciation. (Asahi Television, which is in the same group, uses Japanese readings.)

[...]

As a general rule, Japanese readings [of these names] use kan-on, but in rare instances go-on and kan'yō-on are also used: for example, Kim Il-sung (金日成), Jemulpo (済物浦), and Qian Qichen (銭其琛) are read respectively as Kin Nissei, Saimoppo, and Sen Kishin. [Pure kan-on readings for these would be Kin Jissei, Seibuppo, and Sen Kichin.] Also, it is customary to call Beijing (北京) Pekin, Hong Kong (香港) Honkon, and there are cases where kun-yomi is used, such as Kaohsiung (高雄) in Taiwan, which is called Takao.

The same article also describes the corresponding phenomenon in Korean and Chinese.