Summary: Using the generalized Young inequality, operator convexity and positive operator matrix, we extend and refine some numerical radius inequalities for the weighted sums of Hilbert space operators. Precisely, for \(r, t\in[1, 2]\), if \(T_k, V_k\in\mathscr{B}(\mathscr{H})\) (\(k =1, \dots, n\)) and \(p_k \geqslant 0\) with \(\sum_{k = 1}^n p_k = P_n\), then \[ w^{2r}\left(\frac{1}{P_n}\sum_{k=1}^np_kT_kV_k\right) \leqslant\frac{1}{2^{1/t}}w^{\frac{2}{t}}\left(\frac{1}{P_n}\sum_{k=1}^np_k(|T_k^\ast|^{2rt} + i|V_k|^{2rt})\right) - \inf_{\|x\| = 1}\phi(x), \] where \[ \phi(x) = \frac{1}{4}\left(\left\langle\left(\frac{1}{P_n}\sum_{k=1}^np_k|T_k^\ast|^{2r}\right)x, x\right\rangle - \left\langle\left(\frac{1}{P_n}\sum_{k=1}^np_k|V_k|^{2r}\right)x, x\right\rangle\right)^2. \]