  Considered in this paper is the generalized Camassa-Holm-Novikov equation
with high order nonlinearity, which unifies the Camassa-Holm and Novikov
equations as special cases. We show that the solution map of generalized
Camassa-Holm-Novikov equation is not uniformly continuous on the initial data
in Besov spaces $B_{p, r}^s(\mathbb{R})$ with $s>\max\{1+\frac{1}{p},
\frac{3}{2}\}$, $1\leq p, r< \infty$ as well as in critical space $B_{2,
1}^{\frac{3}{2}}(\mathbb{R}).$ Our result covers and improves the previous work
given by Li et al. \cite{Li 2020, 1Li 2020, Li 2021}(J. Differ. Equ. 269 (2020)
8686-8700; J. Math. Fluid Mech. 22 (2020) 4:50; J. Math. Fluid Mech., (2021)
23:36).
