This is Hayagriva, an incarnation of Vishnu, not to be confused with Dasavatar. See the footnote. The mythology varies. As per Mahabharata, Vishnu as Hayagriva rescued the Vedas from the demons Madhu and Kaitabha. As per Devi Bhagavata, Vishnu fought with and killed a Rakshasa named Hayagriva, assuming the form of Hayagriva avatar.
Here Hayagriva is seated in Ardhapadmasana. His lower right hand is in cinmudra, expounding knowledge. He is holding Shankha, Chakra and Pustaka is his hands, starting from 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 Vedvyasa for 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 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 about 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 or temporary in character and is known as Avesa, while the incarnation of a potion of the power of a divine being is characterize as Amsa.”