  We prove that open discrete mappings of Sobolev classes $W_{\rm loc}^{1, p},$
$p>n-1,$ with locally integrable inner dilatations admit
$ACP_p^{\,-1}$-property, which means that these mappings are absolutely
continuous on almost all preimage paths with respect to $p$-module. In
particular, our results extend the well-known Poletski\u\i\ lemma for
quasiregular mappings. We also establish the upper bounds for $p$-module of
such mappings in terms of integrals depending on the inner dilatations and
arbitrary admissible functions.
