The other answer has already elaborated on the very well known story of Kameshwara and Kamadahana. A very short and precise summary is that Kameshwar Dham, Ballia is said to be the place where Kama was burnt by Lord Shiva by opening his third eye. Kama was sent by Brahma and the other devas in order to disturb Shiva's penance, so that Shiva can bear a child, who was destined to kill the demon Taraka. This is very beautifully described in Chapters 16-19 of Shiva Purana (Rudra Samhita).
I was a bit intrigued by this, and wanted to dig in a bit more about whether Ballia was itself mentioned anywhere.
Starting off from Ramayana, where we actually get the story of Kamadahana. (Tejaswee's answer goes a lot more in detail about this). In Chapter 23 of Balakanda:
Rama and Lakshmana proceeding with Vishvamitra sojourn in a hermitage at a place where River Ganga and River Sarayu are confluent. Once god Shiva with his third eye burnt down the physical entity of Manmatha, the Love-god, at this place.
This is a clear indication, that the actual location of Kamadahana must be somewhere near the confluence of Ganga and Sarayu, which is the River Ghaghra. Ballia is certainly very near to this place, as it is around 50km away (bird distance). It might have potentially been on the confluence itself, as rivers change their courses over time.
Now, jumping into Shiva Purana, Chapter 18 of Rudrasamhita, aptly titled "Description of Perturbation caused by Kama" states:
After going there, the haughty Kama deluded by Siva's magic power, stationed himself, after first spreading the enchanting power of Spring all around.
The enchanting influence of spring spread everywhere around Osadhiprastha, the penance grove of Lord Siva, the supreme Lord, O excellent sage.
O great sage, the groves bloomed with special exuberance, O excellent sage, due to his power.
The fragrant flowers of Mango and Asoka trees shone heightening feelings of love.
Here we get two significant inputs in our quest. First, it mentions about Osadhiprastha as the place where Kama met Siva. Second, it mentions about Mango trees. Cut to the Speaking Tree article cited in the other answer:
The mango tree, which is said to be the same tree as the tree where lord kamdev was hiding still has some burnt areas and still exists
Therefore the presence of Mango trees during Kamadahana is validated in Shiva Purana.
As for Oshadhi Prashta, it was hard to pin point a particular place as I could not find a mention of it in other parts of Shiva Purana (I used the "search in book" option in archive.org). However, a deeper search of that place, returned some interesting results. The book "The Theosophical Glossary"
By Helena Petrovna Blavatsky mentions:
Oshadi Prastha (Sk.). Lit., " the place of medicinal herbs ". A mysterious city in the Himalayas mentioned even from the Vedic period. Tradition shows it as once inhabited by sages, great adepts in the healing art, who used only herbs and plants, as did the ancient Chaldees. The city is mentioned in the Kumara Sambhava of Kalidasa.
This gives us another geographical location about where exactly it might have been located, which is near the Himalayas. While I know that Ballia is in the Gangetic Plains, and not exactly the Himalayas, its proximity to the Himalayas cannot be forgotten.
Therefore to conclude, even though Karon, Ballia hasn't been mentioned, or even referred to in the Puranas, we can infer that Kamadahana happened somewhere near the Himalayas, and somewhere near the confluence of Ganga and Ghagra.
Some additional footnotes that I want to mention:
While searching about the temple itself, I found this other Dainik Bhaskar Hindi News article, which mentions about a certain king Kavaleshwar constructing the temple at Ballia. Searching about King Kavaleshwar did not return any meaningful responses.
I also tried to look for Matsya Purana, where the burning of Kama is written more from a Vishnavaite perspective, but I could not find it online.