This article investigates discrete-time approximations of stochastic integrals driven by semimartingales with jumps via weighted bounded mean oscillation (BMO) approach. This approach enables $L_p$-estimates, $p \in (2, \infty)$, for the approximation error depending on the weight, and it allows a change of the underlying measure which leaves the error estimates unchanged. To take advantage of this approach, we propose a new approximation scheme obtained from a correction for the Riemann approximation based on tracking jumps of the underlying semimartingale. We also discuss a way to optimize the approximation rate by adapting the discretization times to the setting. When the small jump activity of the semimartingale behaves like an $α$-stable process with $α\in (1, 2)$, our scheme achieves under a regular regime the same convergence rate for the error as in Rosenbaum and Tankov [\textit{Ann. Appl. Probab.} \textbf{24} (2014) 1002--1048]. Moreover, our approach extends to the case $α\in (0, 1]$ and to the $L_p$-setting which are not treated there. As an application, we apply the methods in the special case where the semimartingale is an exponential Lévy process to mean-variance hedging of European type options.