No, nothing probably is intact that was written but there is a legend…
from everyone’s favorite Wikipedia
"In Christian mythology, the story of king Abgar of Edessa was an early tale of a wonder-working icon, set in the heart of the region where iconoclast tradition disapproved strongly of images in general, but which this icon-legitimizing legend connected directly with Jesus.
The legend tells that Abgar, king of Edessa, afflicted with an incurable sickness, had heard the fame of the power and miracles of Jesus and wrote to him, acknowledging his divinity, craving his help, and offering him asylum in his own residence; the tradition states that Jesus wrote a letter declining to go, but promising that after his ascension, he would send one of his disciples, endowed with his power, namely Thaddeus (called Addaï), or one of the seventy-two Disciples, called Thaddeus of Edessa.
The 4th century church historian Eusebius, Bishop of Caesarea, records a tradition, in his Historia Ecclesiae, I, xii or xiii, ca AD 325, concerning a correspondence on this occasion, exchanged between Abgar of Edessa and Jesus. Eusebius was convinced that the original letters, written in Syriac, were kept in the archives of Edessa. Eusebius also states that in due course, Judas, son of Thaddaeus, was sent in 29 AD. Eusebius copies the two letters into the text of his history.
The correspondence consisted of Abgar’s letter and the answer dictated by Jesus. As the legend later expanded, a portrait of Jesus painted from life began to be mentioned. This portrait, purportedly painted by the court archivist Hannan during his visit to Jesus, is first mentioned in the Syriac text called the “Doctrine of Addai” (Doctrina Addai-- Addaei, Addaeus = Thaddaeus or Thaddeus), from the second half of the 4th century. Here it is said that the reply of Jesus was given not in writing, but verbally, and that the event took place in 32 AD. This Teaching of Addai is also the earliest account of an image of Jesus painted from life, enshrined by the ailing King Abgar V in one of his palaces. Greek forms of the legend are found in the Acta Thaddaei, the “Acts of Thaddaeus”.
The story of the “letter to Abgar”, including the portrait made by the court painter Hannan, is repeated, with some additions, in the mid-5th century History of the Armenians of Moses of Chorene, who remarked that the portrait was preserved in Edessa.
The story was later elaborated further by the church historian Evagrius, bishop of Edessa (c. 536-600), who declared for the first time (as far as is known) that the image of Jesus was “divinely wrought,” and “not made by human hands.” In sum, the documented legend developed from no image in Eusebius, to an image painted by Hannan in “Addai” and Moses of Chorene, to a miraculously-appearing image not made by human hands in Evagrius."