I think the Skandha Purana story is telling about story of King Sveta. The story is described in Kurma Purana which I also discuss here. I'm just taking relevant verses from that chapter in short. You can read the full story in the above answer:
Who is the king?
The name of that King is Sveta and he was a devotee of Lord Shiva:
श्वेतो नाम शिवे भक्तो राजर्षिप्रवरः पुरा ।
तदाशीस्तन्नमस्कारैः पूजयामास शूलिनम् ॥ ३६.१२
In ancient times, a saintly king, a devotee of Shiva named Sveta, who with the blessings of Siva, used to adore lord Siva, the bearer of the trident.
Who is God Kala?
Here God Kala refers to Yamraja, the God of Death.
सितं कलोजिनं दीप्तं शूलमादाय भीषणम् ।
नेतुमभ्यागतो देशं स राजा यत्र तिष्ठति ॥ ३६.१४
वीक्ष्य राजा भयाविष्टः शूलहस्तं समागतम् ।
कालं कालकरं घोरं भीषणं चण्डदीपितम् ॥ ३६.१५
उबाभ्यामथ हस्ताभ्यां स्पृट्वाऽसौ लिङ्गमैश्वरम् ।
ननाम शिरसा रुद्रं जजाप शतरुद्रियम् ॥ ३६.१६
Thereafter the time of death of the king arrived. In order to lead the king to Yamaloka, the god of death, Kala, arrived, holding a trident in his hand. Then the king Sveta, finding the god of death in the terrific form, holding a trident in his hand, was terrified. Then the king touched Shivalinga with both the hands, bowed in reverence before lord Rudra, reciting the Satarudriya (Sri Rudram) stotra at the same time.
Why was he burnt down?
Kala thought that everyone was in control of him, he says:
तमुवाच भयाविष्टो राजा रुद्रपरायणः ।
एकमीशार्चनरतं विहायान्यन्निषूदय ॥ ३६.१८
इत्युक्तवन्तं भगवानब्रवीद् भीतमानसम् ।
रुद्रार्चनरतो वाऽन्यो मद्वशे को न तिष्ठति ॥ ३६.१९
एवमुक्त्वा स राजानं कालो लोकप्रकालनः ।
बबन्ध पाशै राजाऽपि जजाप शतरुद्रियम् ॥ ३६.२०
The king who was devoted to Rudra, getting terrified, said to Yama, “You leave me since, I am engaged in the worshipping of Shiva, and carry the other people with you and destroy them. Then, lord Kala, spoke to the terrified king. Whether he is engaged in the devotion of Rudra or anyone else, who is not overpowered by me?” Thus speaking, Yama bound the king with his noose, but the king continued to recite the Satarudriya stotra.
Lord Shiva comes there and tells to Kala that the devotee now belongs to him as he is reciting his names and he who remembers him also remembers his form. Kala due to his nature tries to snatch life of Sveta himself and he makes advance towards Lord Shiva also. Lord Shiva kicks Kala with his left foot and kills him:
श्रुत्वा वाख्यं गोपतेरुद्रभावः कालात्माऽसौ मन्यमानः स्वभावम् ।
बद्ध्वा भक्तं पुनरेवाऽथ पाशैः रुद्रो रौद्रमभिदुद्राव वेगात् ॥ ३६.२५
प्रेक्ष्यायान्तं शैलपुत्रीमथेशः सोऽन्वीक्ष्यान्ते विश्वमायाविधिज्ञः ।
सावज्ञं वै वामपादेन कालंत्वेतस्यैनं पश्यतो व्याजघान ॥ ३६.२६
ममार सोऽतिभीषणो महेशपादघातितः ।
रराज देवतापतिः सहोमया पिनाकधृक् ॥ ३६.२७
निरीक्ष्य देवमीश्वरं प्रहृष्टमानसो हरम् ।
ननाम साम्बमव्ययं स राजपुंगवस्तदा ॥ ३६.२८
Listening to the words of lord Shiva, the god of death, because of his nature being horrible, bound the devotee of Shiva in the noose again and in anger he rushed towards lord Shiva. Finding the advance of the god of death, lord Siva, looked at Parvati, the daughter of the mountain, and kicked him with his left foot, while king Sveta was looking on. With the kicking by lord Siva, the awful Kaladeva was killed and the Pinaka bearing lord Siva looked graceful with the goddess Uma. The king Sveta felt delighted at the Sight of lord Siva Offering his salutation to Siva, he started eulogising him.
Since, he was angry his Teja (lusture) while kicking Kala should have burned the Kala and killed him. Latter on the request of Lord Brahma, Kaladeva was again revived by Lord Shiva.
Edit:
As described by Swiftpushkar in his answer, this event with King Sveta is itself mentioned in Skandha Purana also where it is mentioned that Lord Shiva burned King Sveta with his third eye. Since events frequently repeat in cosmic timescales and Puranas tell story of different Kalpas in some incidents Kala was kicked and killed while in some he was burned by opening of the third eye. Quoting from the same answer:
On hearing Yama's words , Kala became infuriated. Hurriedly he took up the sword intending to kill the king. The king is thinking about the foot of Shiva who is of the nature of supreme being , and the cause of salvation and complete identity on form . Kala was rushing at him and was seen by Sadashiva , the slayer of Kala. The rouge was approching him undeterred arrogantly. When he was seen midway between himself and nandikeshwara by shiva , the lord of the universe , the affectionate kinsman of devotee , he was stared at with the Third eye by the great lord who wanted to protect his devotee. He became reduced to ash in an instant. The king then regained conciniousness and lookedagain and again at kala who had come to kill him and then was burnt by fire. Without being excited , who prayed to Rudra who resembled the fire of kala. (Skandha Purana, Maheswara Khanda, chapter 32)
