  The lack of rotating black holes, typically found in nature, hinders testing
modified gravity from astrophysical observations. We present the axially
symmetric counterpart of an existing spherical hairy black hole in Horndeski
gravity having an additional deviation parameter $Q$, which encompasses the
Kerr black hole as a particular case ($Q=0$). We investigate the effect of
Horndeski parameter $Q$ on the rotating black holes' geometry and analytically
deduce the gravitational deflection angle of light in the weak-field limit. For
the S2 source star, the deflection angle for the Sgr A* model of the rotating
Horndeski gravity black hole for both prograde and retrograde photons is larger
than the Kerr black hole values. We show how parameter $Q$ could be constrained
by astrophysical implications of the lensing of this object. The thermodynamic
quantities, Komar mass, and Komar angular momentum gets corrected by the
parameter $Q$, but the Smarr relation $M_{\text{eff}}=2ST+2\Omega
J_{\text{eff}}$ still holds at the event horizon.
