  Let $a(n)$ be the Fourier coefficients of a holomorphic cusp form of weight
$\kappa=2n\geqslant12$ for the full modular group and
$A(x)=\sum\limits_{n\leqslant x}a(n)$. In this paper, we establish an
asymptotic formula of the fourth power moment of $A(x)$ and prove that
\begin{equation*}
  \int_1^TA^4(x)\mathrm{d}x=\frac{3}{64\kappa\pi^4}s_{4;2}(\tilde{a})
T^{2\kappa}+O\big(T^{2\kappa-\delta_4+\varepsilon}\big) \end{equation*} with
$\delta_4=1/8$, which improves the previous result.
