  In this paper, we are concerned with the estimates for the moments of
stochastic convolution integrals. We first deal with the stochastic singular
integral operators and we aim to derive the Morrey-Campanato estimates for the
$p$-moments (for $p\ge1$). Then, by utilising the embedding theory between the
Campanato space and H\"older space, we establish the norm of
$C^{\theta,\theta/2}(\bar D)$, where $\theta\ge0, \bar D=\bar G\times[0,T]$ for
arbitrarily fixed $T\in(0,\infty)$ and $G\subset\mathbb{R}^d$. As an
application, we consider the following stochastic (fractional) heat equations
with additive noises
  \bess du_t(x)=\Delta^\alpha u_t(x)dt+g(t,x)d\eta_t,\ \ \ u_0=0,\ 0\leq t\leq
T, x\in G,
  \eess where $\Delta^\alpha=-(-\Delta)^\alpha$ with $0<\alpha\leq1$ (the
fractional Laplacian), $g:[0,T]\times G\times\Omega\to\mathbb{R}$ is a joint
measurable coefficient, and $\eta_t, t\in[0,T]$, is either the Brownian motion
or a L\'evy process on a given filtered probability space
$(\Omega,\mathcal{F},P;\{\mathcal{F}_t\}_{t\in[0,T]})$. The Schauder estimate
for the $p$-moments of the solution of the above equation is obtained. The
novelty of the present paper is that we obtain the Schauder estimate for
parabolic stochastic partial differential equations with L\'evy noise.
