  In the present paper we investigate the $L_1$-weak ergodicity of
nonhomogeneous discrete Markov processes with general state spaces. Note that
the $L_1$-weak ergodicity is weaker than well-known weak ergodicity. We provide
a necessary and sufficient condition for such processes to satisfy the
$L_1$-weak ergodicity. Moreover, we apply the obtained results to establish
$L_1$-weak ergodicity of discrete time quadratic stochastic processes. As an
application of the main result, certain concrete examples are also provided.
