Let $L\subset C$ be a regular Jordan curve. In this work, the approximation properties of the $p$-Faber-Laurent rational series expansions in the $\omega $ weighted Lebesgue spaces $L^p(L,\omega )$ are studied. Under some restrictive conditions upon the weight functions the degree of this approximation by a $k$th integral modulus of continuity in $L^p(L,\omega )$ spaces is estimated.