  In this paper, we consider a kind of degenerate normalized $ p $-Laplacian
equation with variable exponents for $ 1 <p<\infty$. We establish local $
C^{1,\alpha'} $ regularity of viscosity solutions by making use of the
compactness argument, scaling techniques and the localized oscillating method.
In addition, we also obtain almost optimal pointwise $ C^{1,\tau} $ regularity
for a new model of the normalized $ p$-Laplacian equation involving
non-homogeneous degenerate term $H(x,Du)$. The method in this paper is based on
an improved oscillation-type estimate inspired by the ideas in Attouchi et al.
(J. Math. Pures Appl, \textbf{108}: 553-591, 2017), which is different from the
approach in the recent work by Jesus (Calc. Var. Partial Differential
Equations, \textbf{61} (2022), Paper No. 29, 21 pp).
