Shiva is called as Ayonija, one who does not stay in the womb.
Nor does He have children in the sense we understand it or even like the Avatars of Vishnu.
However there are references in the Puranas that Shiva had children.
They are Eight.
This does not include the refernces found in the Kandha Puranam about the Nava Veeras who were born with Subrahmanya
However they were not because of the Union between Shiva and Shakthi or even a woman.
- Ganesha -Born of Shakthi’s sweat.
- Kartikaya, Subrahmanya-Of Shiva’s Third Eye.
- Ashok Sundari -Of Parvati’s thought of Shiva
Lord Shiva being of Yogic disposition, goes off to Kailash often.
Feeling lonely Parvati prayed the Kalpa Vruksha for a companion and thus was born Ashoka Sunadri.(Shiva Purana)
She is considered as the child of Shiva along with Subrahmanya and Ganesha.
When Shiva was meditating on Mandara mountain, Parvati who was in a playful mood covered Shiva’s eyes due to which the whole universe was covered in darkness. The sweat that oozed out of Parvati’s hands due to touching Shiva fell to the ground and created a horrible looking and blind boy. Parvati was terrified on seeing him however Shiva said that since he was born due to their physical contact he was their child. When the demon king Hiranyaksha who was childless performed penance to please Shiva in order to beget a child Shiva gifted the child to him and named him Andhaka due to his blindness. After Hiranyaksha’s death Andhaka became the king however he was not regarded as an Asura since he was a divine product. Disowned by majority of his clan he performed a severe penance to please Brahma. Brahma thus appeared to him and offered him a boon. Andhaka demanded Brahma to make him invincible and to repair his vision. Brahma granted these wishes however warned him that he can be killed by Shiva. Andhaka went back to his kingdom and subdued all his opponents and even the Devas
When Indra and Brihaspati were going towards Mount Kailash to meet Shiva their way was blocked by a naked yogi with matted hair and a radiant face. The yogi was Shiva himself, who had taken the form to test the knowledge of Indra and Brihaspati. Indra did not recognize the yogi and was infuriated at the fact that the man was not moving out of their way. Indra asked him to move but the man did not budge. After getting no reply Indra became enraged and threatened him with his thunderbolt. Upon this action Indra’s arm became paralyzed and Shiva neutralized the thunderbolt. Shiva became angry upon this action of Indra and his eyes turned red, frightening Indra. The anger caused Shiva’s third eye to open, nearly killing Indra. Brihaspati recognized Shiva and prayed to him, requesting him to pardon Indra. To avoid killing Indra, Shiva sent the fire from his eye towards the ocean and upon meeting with the ocean it assumed the form of a boy.The boy cried terribly which caused Brahma to descend from heaven. The ocean told Brahma that he did not know where the boy came from. Brahma then told him that the boy will one day become the emperor of Asuras, he could only be killed by Shiva and after his death he would return to Shiva’s third eye.
7.Manasa, Snake Goddess.
By the 14th century, Manasa was identified as the goddess of fertility and marriage rites and was assimilated into the Shaiva pantheon as a relative of Shiva. Myths glorified her by describing that she saved Shiva after he drank poison, and venerated her as the “remover of poison”. Her popularity grew and spread to southern India, and her cult began to rival Shaivism itself. As a consequence, stories attributing Manasa’s birth to Shiva emerged and ultimately Shaivism adopted this indigenous goddess into the Brahmanical tradition of mainstream Hinduism.
8.JyothiJyoti, the goddess of light, who emerges from Shiva’s halo and is the physical manifestation of his grace. She is commonly associated with Kartikeya.
References from Shiva Purana .
For Jothy it is a Legend. I do not find any reference int the Eighteen Puranas.