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The Wikipedia page Extreme weather events of 535–536 lists under "Documentary evidence" that:

• Low temperatures, even snow during the summer (snow reportedly fell in August in China during the Northern and Southern dynasties, which
  caused the harvest there to be delayed


• "A dense, dry fog" in the Middle East, China and Europe

The page does provide a reference (Ochoa, George; Jennifer Hoffman; Tina Tin (2005). Climate: The Force That Shapes Our World and the Future of Life on Earth. Emmaus, PA: Rodale. p. 71.), but upon reading the referenced book, I noticed that the original quote was itself unsubstantiated, as shown in the picture I took below (at the end of the book there was only a Further Reading list):

I then proceeded to find other academic sources online. One article titled Spatial Response to Major Volcanic Events in or about AD 536, 934 and 1258 includes a passage quite similar to that found on the Wikipedia page, except slight variation in time:

The above results for Mongolia are consistent with the observations of
summer frosts and snow, crop failure, decreased star visibility and
famine documented for northern China in July and August of AD 536–7.

it listed the following citation for that passage:

Pang, K. and Chou, H.-H.: 1985, Unpublished research, in S. Weisbard,
‘Excavating Words: A Geological Tool’, Science News 127, 91–94.

So I went to this cited article, which contains the following:

The effect of this volcano was also felt in China. Pang reports that,
during the spring and fall equinoxes each year, the ancient Chinese
looked for Canopus -the brightest star in the constellation of Alpha
Carina-to assure themselves of good times ahead and to demark the
seasons. In A.D. 536, however, Canopus was not seen. Pang found
records from the state of Ching in southern China reporting frost and
snow in July and August that killed the seedling crop, causing a major
famine the following autumn. The accounts also show that the effects
of the eruption were noted until 538. Other kingdoms reported similar
disasters; one record indicated that the weather was so severe that 70
to 80 percent of the people starved to death, says Pang.

Unfortuantely, the article does not provide any specific references to this quote, and it seems that Pang's research was indeed unpublished.

I have just begun to look into the massive amount of Chinese chronicles from this period with the hope of finding the exact entries quoted by Pang, in particular the highlighted quote concerning the severity of the disaster, which seems like to be highly exaggerated. In the meantime, I would really appreciate if someone could point me to a specific passage from an ancient Chinese source that confirms aforementioned descriptions.

Thank you.

The chronicles Beishi "History of the North" and Nanshi "History of the South" have at least some of what you are looking for.

The Nanshi chronicle mentions frost and snow. This is cited in Catastrophe: An Investigation into the Origins of the Modern World by David Keys:

In July 537 China was hit by frost, while in August it snowed. The
History of the Southern Dynasties recorded that “in July in Qingzhou
and [another province] there was a fall of frost” and that “in August
in Qingzhou there was snow,” which “ruined the crops.”

The Nanshi chronicle is also cited in the The Palgrave Handbook of Climate History:

the early seventh-century Nanshi chronicle refers to “yellow dust”
that “fell like snow” in 536 and 537. In the latter year, it “filled
scoops when picked up.” The dust was almost certainly Gobi sand (not
volcanic ash), but this signals that 536 and 537 were unusually dry.

Keys also has the following paragraph which cites the Beishi:

“Because of drought, there was an imperial edict which ordered that in
the capital [Ch’ang-an], in all provinces, commanderies and districts,
one should bury the corpses,” says the Bei shi for late April to early
May 535.1 “[There was] great drought. [The government] had to provide
water at the city gates [of Ch’ang-an] and the hall gates [of the
palace] as well as the gates of the government offices,” says the
entry for late June and early July of the same year. Then in September
536, in the north Chinese “provinces of Bian, Si, Zhuo and Jian, hail
fell” and there was “a great famine.” By December the situation had
deteriorated to such an extent that the government had to send special
inspectors “to investigate [the conditions of] the famished refugees
who were roaming around north of the Yellow River.”

The next paragraph then adds:

And in Shaanxi province, “the land within the Passes,” The Annals of
the Western Wei in the Bei shi state that there was “a great famine,”
and that “the people practiced cannibalism and 70 to 80 percent of the
population died.”

This is the supplementary answer to that of Lars Bosteen.

As he said, according to this site( Japanese ), Nanshi records multiple quite unusual events occurred in Lyan Dynasty ( English ).

Quote ( from the above Japanese source )

534年閏12月丙午、西南方で雷鳴が二度、響いた。

In the intercalary December, 534AD, we heard the lightening sound twice at South West. ( Nanshi 1.7.4.1 )

これは先にも引用した火山噴火への言及だが、恐らく梁の人々はこのようなことはすぐ忘れてしまっただろう。不可思議なことと言ってもやはり、庶民には何の関係もない･･･筈だった。しかしそれから異常なことが次々と起こった。翌535年の冬十月、雪のような黄塵が降った1。これが火山灰だったか気候の大異変に起因する季節はずれの砂嵐だったかは今となっては確かめる術もない。この黄塵は翌536年11月2、537年2月1日3、550年1月丁巳の日[4]にも観測された。

Gradually updated due to the enormous difficulties of the translation.

Conclusion

Nanshi, the chronicle of events of the Lyan Dynasty, records multiple events of unusual climate events that occurred from the 534AD to 550AD, as also the 北史 ( I am sorry I can not translate ), the chronicle of events of the Northern Wei Dynasty records the same very unusual climate events which occurred around the same period that were recorded by Nanshi.