Summary: Let \(\{X_{nk},\,u_n\leq k\leq v_n,\,n\geq 1\}\) and \(\{A_{nk},\,u_n\leq k\leq v_n,\,n\geq 1\}\) be two arrays of random variables defined on the same probability space \((\Omega,\mathcal A,P)\) and \(\mathcal B_n\) be sub-\(\sigma\)-algebras of \(\mathcal A\). Let \(r>0\) be a constant. In this paper, we introduce some concepts of conditional residual \(h\)-integrability such as conditionally residually \(h\)-integrable relative to \(\mathcal B_n\) concerning the array \(\{A_{nk}\}\) with exponent \(r\) and conditionally strongly residually \(h\)-integrable relative to \(\mathcal B_n\) concerning the array \(\{A_{nk}\}\) with exponent \(r\). These concepts are more general than some known settings of randomly weighted sums of random variables. Based on the conditions of conditional residual \(h\)-integrability with exponent \(r\) and conditional strongly residual \(h\)-integrability with exponent \(r\), we obtain the conditional mean convergence and conditional almost sure convergence for randomly weighted sums.