  Background: The discovery of neutrinoless double-beta ($0\nu\beta\beta$)
decay would demonstrate the nature of neutrinos, have profound implications for
our understanding of matter-antimatter mystery, and solve the mass hierarchy
problem of neutrinos. The calculations for the nuclear matrix elements
$M^{0\nu}$ of $0\nu\beta\beta$ decay are crucial for the interpretation of this
process. Purpose: We study the effects of relativity and nucleon-nucleon
short-range correlations on the nuclear matrix elements $M^{0\nu}$ by assuming
the mechanism of exchanging light or heavy neutrinos for the $0\nu\beta\beta$
decay. Methods: The nuclear matrix elements $M^{0\nu}$ are calculated within
the framework of covariant density functional theory, where the
beyond-mean-field correlations are included in the nuclear wave functions by
configuration mixing of both angular-momentum and particle-number projected
quadrupole deformed mean-field states. Results: The nuclear matrix elements
$M^{0\nu}$ are obtained for ten $0\nu\beta\beta$-decay candidate nuclei. The
impact of relativity is illustrated by adopting relativistic or nonrelativistic
decay operators. The effects of short-range correlations are evaluated.
Conclusions: The effects of relativity and short-range correlations play an
important role in the mechanism of exchanging heavy neutrinos though the
influences are marginal for light neutrinos. Combining the nuclear matrix
elements $M^{0\nu}$ with the observed lower limits on the
$0\nu\beta\beta$-decay half-lives, the predicted strongest limits on the
effective masses are $|\langle m_\nu\rangle|<0.06~\mathrm{eV}$ for light
neutrinos and $|\langle m_{\nu_h}^{-1}\rangle|^{-1}>3.065\times
10^8~\mathrm{GeV}$ for heavy neutrinos.
