This is Hayagriva, an incarnation of Vishnu, not to be confused with Dasavatara. See the footnote.  The mythology varies. As per the Mahabharata,  Vishnu,  as Hayagriva, rescued the Vedas from the demons Madhu and Kaitabha. As per the Devi Bhagavata, Vishnu fought with and killed a Rakshasa named Hayagriva, assuming the form of the Hayagriva avatar.

However, Hayagriva, with a horse head and the human body, is considered one of the three major Hindu deities of learning. The other two are Saraswati and Dakshinamurthy.

Here, Hayagriva is seated in Ardhapadmasana. His lower right hand is in cinmudra, expounding knowledge. He is holding Shankha, Chakra and Pustaka in his hands, starting from the upper right hand clockwise. The attributes seem to vary. Please see another example of Hayagriva, in this blog site, where he is holding Akshamala instead of Chakra.

In this example, Hayagriva is sitting on a tortoise. See Vedavyasa for a similar asana.

The prabhavali does not belong to this Vigraha and belongs to a Saivite deity. See the Trishul and Drum on the prabhavali, in line with the Cobra hood and on the outer edge of the Prabhavali. I retained the prabhavali, as it is my practice to leave Vigrahas as I got them, except for cleaning when it is necessary.

This idol is from  North Karnataka or nearby areas. The Vigraha is about 10.5 cm in height, and with Prabhavali, the height is 16.5 cm.  The idol shows significant ritual wear. This may not date earlier than the eighteenth century.

Incarnations of Vishnu:  T A G Rao, in his book on Elements of Hindu Iconography, mentions Vishnu’s incarnations are of three kinds (1) Avatara (2) Avesa and (3) Amsa. “A complete incarnation is designated as an Avatar, a partial incarnation is more temporary in character and is known as Avesa, while the incarnation of a portion of the power of a divine being is characterised as Amsa.”