For ease of comparison, most Japanese Kanji text in this answer will be rendered in Kyūjitai, which are almost 100% identical to Korean Hanja.
Short answer
No, Korean mixed script (「[國漢文混用]{국한문혼용}」, 漢字ハングル混じり文) does not employ the equivalent of Japanese kun'yomi (「[訓讀]{훈독}」, hun-dok).
Longer answer
They do exist, but you wouldn't come across any of significance if you were beginning to learn Korean in mixed script. Korean actually doesn't classify these as such anyway, as all Hanja readings for vocabulary that can be written in Hanja are classified as on'yomi ([音讀]{음독}, eum-dok) by definition, while Japanese would sometimes classify these as other readings. These universally come from three sources, which are:
Historical misreadings. For example, the counter「個」(equivalent to Japanese usage) has a reading of「개」(gae), which is really the reading of「介」; this misreading came from the confusion of「介」with「个」, which was a historical shorthand version of「個」. The proper 音讀 reading that fulfils Middle Chinese to Korean sound shifts would be「거」(geo), but this is not the standard reading in dictionaries so cannot be considered as 音讀.
The Korean equivalent of ateji, in the sense of phonetic loans for unrelated words. For example,「[寸]{촌}」(chon) is used throughout East Asia as a unit of distance measurement, while in Korean it has an additional meaning of degree of kinship;「[三寸]{삼촌}」(sam-chon) means uncle.
Phonological corruptions. The origins of this kind of vocabulary is shrouded in mystery, and may or may not be Chinese or "Korean"* at all, attested in extremely old proper nouns; for example, there are various hypotheses of the origin of two of the names of Korea,「朝鮮」and「韓」, both speculated to be of non-Chinese in origin.
- Note that Japanese has this type of corruption too; 「うま」and「うめ」, although treated as kun'yomi, are almost definitely cognate to Chinese「馬」and「梅」.
Regardless, there is one rule that is always applied to Hanja without exception, and that is each Hanja is always one Hangeul syllable block long.
There is also the far more numerous case of Chinese-originated vocabulary in Korean (「[漢字語]{한자어}」) not being rendered in Hanja, which makes them anything but on'yomi . These are also phonological corruptions. For example,「[김치]{キムチ}」(kimchi) comes from the Hanja「[沈菜]{침채}」(chim-chae, meaning soaked vegetables), but is not written or pronounced this way anymore.
音讀 and 訓讀 are exactly analogous between Korean and Japanese dictionaries for 漢字 entries. In both, 音讀 is treated as largely a Chinese-originated sound gloss, while 訓讀 is treated as largely** a native-originated meaning gloss. However, Korean 訓讀 does not extend outside of dictionaries, unlike kun'yomi; only 音讀 can be written in Hanja, while 訓讀 can never be written in Hanja.
This also means that the misreadings and phonetic loans, and phonological corruptions mentioned above are treated as 音讀, and all of the Chinese-originated vocabulary not rendered in Hanja are not treated as any kind of 讀 for Hanja at all.
Lastly, as in Japanese, Korean spoken language and written language differ in vocabulary usage; the former tends to use more native vocabulary while the latter has a higher degree of 漢字語.
- This means that sometimes when reading off a piece of text with Hanja, 訓讀 may be used; for example,「[二時]{이시}」(i-shi, same meaning as Japanese) may be read aloud as「두시」(du-shi), where「두」is the native Korean number for two. However, this probably shouldn't be considered as a proper way of reading Hanja.
- This also means that, as the degree of technicality or formality increases in a piece of text, Korean and Japanese greatly converge in form. See, for example, the following Sōshi-kaimei notice during the Japanese occupation of Korea, where it was only necessary to convert between kana and hangul, keeping all 漢字 the same:
*Quotes are put here because, unlike Chinese, the origin of the Korean (and Japanese) languages is extremely uncertain from ancient times.
**Kun'yomi, even minus the readings from borrowings from western languages, is not exclusively restricted to native Japanese vocabulary. There are instances of Kanji where it is treated either as if (1) the kun'yomi does not exist, or (2) the kun'yomi is exactly the same as the on'yomi. A common example would be「茶」.
