  We consider a class of variable-exponent mixed fully nonlinear local and
nonlocal degenerate elliptic equations, which degenerate along the set of
critical points, $C:=\big\{x:\,Du(x)=0\big\}.$ Under general conditions, first,
we establish the Lipschitz regularity of solutions using the Ishii-Lions
viscosity method when the order of the fractional Laplacian,
$s\in\big(\frac{1}{2},1\big).$ Due to inapplicability of comparison principle
for the equations under consideration, one can not use the classical Perron's
method for the existence of a solution. However, using the Lipschitz estimates
established in theorem and vanishing viscosity method, we get the existence of
solution. We further prove interior $C^{1,\delta}$ regularity of the viscosity
solutions using an improvement of the flatness technique when $s$ is close
enough to $1.$
