In the chapter 20, Section 3 of Satarudra–Samhita of Shiva Purana, the incarnation of Shiva as Hanumat is mentioned.
Shiva gave blessings to give birth to a son with his power to Anjani, the daughter of Gautama, for Rama’s work. In due course Shiva was born of it in the form of a monkey named Hanumat to Anjani & her husband Kesari. He had great strength and exploit.
Early in the morning he took the disc of the sun for a small fruit and wanted to eat it. At the request of the gods, he left it. Knowing him to be an incarnation of Shiva of great power the gods and the sages granted him boons. At her behest the bold monkey went to the sun every day and learnt all lore from the sun without any strain.
The excellent monkey, a part of Shiva approached Sugriva, a part of the sun, at his behest, after getting the permission of his mother. Sugriva who had been exiled by his elder brother Bali who enjoyed his (Sugriva’s) wife, stayed on the mountain Ṛṣyamuka long with Hanumat. The intelligent monkey became Sugriva’s minister and did everything beneficial to him in every respect. He made him enter into a pleasing alliance with Rama who came there along with his brother and who was sad because his wife had been abducted by Ravaṇa.
Rama slew the powerful monkey Bali who was a sinner as he enjoyed his brother’s wife and who professed to be a great hero. Then at the behest of Rama, Hanumat went out in search of Sita along with many monkeys. On knowing that she was in Lanka. the leading monkey quickly jumped across the ocean, never before crossed by any and reached Lanka. There he performed wonderful exploits and handed over the excellent token of recognition of his lord Rama to Sita. He dispelled the sorrow of Sita. Taking a token from her he turned back to go towards Rama. On the way he broke Ravaṇa’s park and killed many Rakṣasas. At the same time he killed Ravaṇa’s son also along with many Rākṣasas. That fearless monkey of great exploits wrought a havoc there. When the powerful Ravaṇa bound him with oil-dripping cloths and set fire to them, the monkey, a part of Shiva himself, jumped about and making that a pretext burnt the city of Lanka. After burning Lanka except the mansion of Vibhiṣaṇa, the heroic leader of monkeys jumped into the ocean to extinguish the fire.
Then Hanumat reached Rama’s side without delay and handed over the crest-jewel of Sita. At his behest, monkeys gathered many mountains and built a bridge in the ocean.
At the time when he was about to cross the ocean, Rama installed a phallic image of Shiva and worshipped it with a desire to be victorious. Securing his boon of victory, he crossed the ocean, surrounded Lanka with the monkeys and fought against the Rakṣasas. The heroic Hanumat slew many Asuras and protected Rama’s army. He enlivened Lakṣmaṇa by the medicinal herb Sanjivani when he was wounded by Meghanada’s javelin. He slew the demon Ahiravaṇa and brought Rama and Lakṣmaṇa from his place to their own after guarding them well.
Hanumat quickly carried out the tasks of Rama by all means, made the Asuras bow down. Himself a leader of devotees, he made Ma Sita and Rama happy. He established the cult of devotion to Rama in the world.
Indeed, he was the Incarnation of Shiva the lord and uplifter of devotees. He was the life-saviour of Lakṣmaṇa and the destroyer of the arrogance of all Daityas. He was the heroic Hanumat who carried the tasks of Rama always. In the world he is called Rama’s messenger. He is favourably disposed to his devotees.