  The action of four-dimensional Horndeski gravity coupled to Born-Infeld
electromagnetic fields is given via the Kaluza-Klein process. Dyonic black hole
solution of the theory is constructed. The metric is devoid of singularity at
the origin independent of the parameter selections, this property is different
from the one of Einstein-Born-Infeld black holes. Thermodynamics of the black
hole is studied, thermodynamic quantities are calculated and the first law is
checked to be satisfied. Thermodynamic phase transitions of the black holes are
studied in extended phase space.
