  In this paper, we investigate the explicit birational geometry for projective
$\epsilon$-lc varieties polarised by nef and big Weil divisors. We show that if
$X$ is a projective $\epsilon$-lc variety, $H$ is a nef and big Weil divisor
with $\dim\overline{\varphi_{H}(X)}\geq n-1$ and $L$ is an effective Weil
divisor such that $|L-K_X|\neq \emptyset$ or $L-K_X$ is nef, then we can find
an explicit lower bound of $\text{vol}(H)$ and prove that $|L+m^\prime H|$ is
birational for $m^\prime\geq m$, where $m$ is an explicit number which depends
only on $n$ and $\epsilon$. This result can be applied to polarised Calabi-Yau
varieties, Fano varieties and varieties of general type, generalizing the
results in [CEW22] and [Zhu23].
