  Let $G$ be a graph with $m$ edges and let $\mathrm{mc}(G)$ denote the size of
a largest cut of $G$. The difference $\mathrm{mc}(G)-m/2$ is called the surplus
$\mathrm{sp}(G)$ of $G$. A fundamental problem in MaxCut is to determine
$\mathrm{sp}(G)$ for $G$ without specific structure, and the degree sequence
$d_1,\ldots,d_n$ of $G$ plays a key role in getting lower bounds of
$\mathrm{sp}(G)$. A classical example, given by Shearer, is that
$\mathrm{sp}(G)=\Omega(\sum_{i=1}^n\sqrt d_i)$ for triangle-free graphs $G$,
implying that $\mathrm{sp}(G)=\Omega(m^{3/4})$. It was extended to graphs with
sparse neighborhoods by Alon, Krivelevich and Sudakov. In this paper, we
establish a novel and stronger result for a more general family of graphs with
sparse neighborhoods.
  Our result can derive many well-known bounds on surplus of $H$-free graphs
for different $H$, such as triangles, even cycles, graphs having a vertex whose
removal makes them acyclic, or complete bipartite graphs $K_{s,t}$ with $s\in
\{2,3\}$. It can also deduce many new (tight) bounds on $\mathrm{sp}(G)$ in
$H$-free graphs $G$ when $H$ is any graph having a vertex whose removal results
in a bipartite graph with relatively small Tur\'{a}n number, especially the
even wheel. This contributes to a conjecture raised by Alon, Krivelevich and
Sudakov. Moreover, we obtain new families of graphs $H$ such that
$\mathrm{sp}(G)=\Omega(m^{3/4+\epsilon(H)})$ for some constant $\epsilon(H)>0$
in $H$-free graphs $G$, giving evidences to a conjecture suggested by Alon,
Bollob\'as, Krivelevich and Sudakov.
