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The Acts of Thaddeus recounts the legend of King Abgar V being healed by a cloth with the image of Jesus on it.
The Acts of Thaddeus (Greek: Πραξεὶ̀ς τοῦ Θαδδαίου) is a Greek document written between 544 and 944 CE which purports to describe correspondence between King Abgar V of Edessa and Jesus, which results in Jesus' disciple Thaddeus going to Edessa.
Passage that refers to the image on the cloth:
And Ananias, having gone and given the letter, was carefully looking at Christ, but was unable to fix Him in his mind. And He knew as knowing the heart, and asked to wash Himself; and a towel was given Him; and when He had washed Himself, He wiped His face with it. And His image having been imprinted upon the linen, He gave it to Ananias, saying: Give this, and take back this message, to him that sent thee: Peace to thee and thy city! For because of this I am come, to suffer for the world, and to rise again, and to raise up the forefathers. And after I have been taken up into the heavens I shall send thee my disciple Thaddæus, who shall enlighten thee, and guide thee into all the truth, both thee and thy city.
And having received Ananias, and fallen down and adored the likeness, Abgarus was cured of his disease before Thaddæus came.
In the passage, "towel" is "tetradiplon" in the Greek.
There is an earlier account of the legend of Abgar in the Doctrine of Addai.
The Doctrine of Addai (Syriac: ܡܠܦܢܘܬܐ ܕܐܕܝ ܫܠܝܚܐ Malp̄ānūṯā d-Addai Šlīḥā) is a Syriac Christian text, written in the late 4th or early 5th century CE. It recounts the legend of the Image of Edessa as well as the legendary works of Addai and his disciple Mari in Mesopotamia.
In this legend, the image was painted.
https://web.archive.org/web/20170904034 ... 2_text.htm
When Hannan, the keeper of the archives, saw that Jesus spake thus to him, by virtue of being the king's painter, he took and painted a likeness of Jesus with choice paints, and brought with him to Abgar the king, his master. And when Abgar the king saw the likeness, he received it with great joy, and placed it with great honour in one of his palatial houses.
At the moment that he placed his hand upon him, he was cured of the plague of the disease, which he had had for a long time. Abgar wondered and was astonished, that as it was reported to him concerning Jesus, that which He did and cured; so also Addai himself, without medicine of any kind, healed in the name of Jesus.
Both of these accounts were written hundreds of years after the life of Abgar V, who lived in the first century.
Abgar V (c. 1st century BC – c. AD 50), called Ukkāmā (meaning "the Black" in Syriac and other dialects of Aramaic),[a] was the King of Osroene with his capital at Edessa.
Abgar V is said to be one of the first Christian kings in history, having been converted to the faith by Thaddeus of Edessa, one of the seventy disciples.
In the apocryphal work, The Epistles Of Jesus Christ & Abgarus King Of Edessa, it does not mention any cloth or painting.
https://en.wikisource.org/wiki/The_supp ... /Chapter_6
The answer of Jesus by Ananias the footman to Abgarus the king, 3 declining to visit Edessa.
ABGARUS, you are happy, forasmuch as you have believed on me, whom you have not seen.
2 For it is written concerning me, that those who have seen me should not believe on me, that they who have not seen might believe and live.
3 As to that part of your letter, which relates to my giving you a visit, I must inform you, that I must fulfil all the ends of my mission in this country, and after that be received up again to him who sent me.
4 But after my ascension I will send one of my disciples, who will cure your disease, and give life to you, and all that are with you.
New Advent article does not describe King Abgar's healing by the cloth, but by the laying on of hands.
After the ascension of our Saviour, the Apostle Thomas, one of the twelve, sent one of the seventy-six disciples, Thaddæus, to the city of Edessa to heal Abgar and to preach the Gospel, according to the word of the Lord. Thaddæus came to the house of Tobias, a Jewish prince, who is said to have been of the race of the Pacradouni. Tobias, having left Archam, did not abjure Judaism with the rest of his relatives, but followed its laws up to the moment when he believed in Christ. Soon the name of Thaddæus spreads through the whole town. Abgar, on learning of his arrival, said: "This is indeed he concerning whom Jesus wrote to me;" and immediately Abgar sent for the apostle. When Thaddæus entered, a marvellous appearance presented itself to the eyes of Abgar in the countenance of the apostle; the king having risen from his throne, fell on his face to the earth, and prostrated himself before Thaddæus. This spectacle greatly surprised all the princes who were present, for they were ignorant of the fact of the vision. "Are you really," said Abgar to Thaddæus, "are you the disciple of the ever-blessed Jesus? Are you he whom He promised to send to me, and can you heal my maladies?" "Yes," answered Thaddæus; "if you believe in Jesus Christ, the Son of God, the desires of your heart shall be granted." "I have believed in Jesus," said Abgar, "I have believed in His Father; therefore I wished to go at the head of my troops to destroy the Jews who have crucified Jesus, had I not been prevented by reason of the power of the Romans."
Thenceforth Thaddæus began to preach the Gospel to the king and his town; laying his hands upon Abgar, he cured him; he cured also a man with gout, Abdu, a prince of the town, much honoured in all the king's house. He also healed all the sick and infirm people in the town, and all believed in Jesus Christ. Abgar was baptized, and all the town with him, and the temples of the false gods were closed, and all the statues of idols that were placed on the altars and columns were hidden by being covered with reeds. Abgar did not compel any one to embrace the faith yet from day to day the number of the believers was multiplied.
King Abgar is a saint in the Orthodox church with several feasts in his honor.
Abgar is counted as saint, with feasts on 11 May and 28 October in the Eastern Orthodox Church, Thursday of the Third Week of Lent (Mid-Lent) in the Syriac Orthodox Church, and daily in the Mass of the Armenian Apostolic Church.