The study of $\beta$-delayed decay of nuclei near the proton drip line provides a powerful tool to understand the role of isospin-symmetry breaking in the structure of proton-rich nuclei. A $\beta$-delayed process involves first a $\beta$-decay of a precursor, with a large superallowed branch populating the isobaric analogue state (IAS), followed by emission of charged particles (protons, diprotons, alpha particles, clusters) or gamma radiation. The typical $Q$ value systematics of these decays is such that the second-stage proton (or multi-particle) emission from the IAS is isospin-forbidden, whereas decay from Gamow-Teller populated states is consistent with the isospin-symmetry limit. The experimental data on isospin-forbidden proton-emission branching ratios provides a stringent test for charge-dependent terms of the nuclear Hamiltonian. In this contribution, we present a shell-model study of the partial-decay schemes of some recently measured very neutron deficient silicone isotopes, e.g., $^{23}$Si, $^{24}$Si, $^{25}$Si, as well as a $pf$-shell precursor, $^{53}$Ni. We use a microscopic isospin-nonconserving (INC) Hamiltonian which allows us to account for the isospin-symmetry breaking consistently in all physics processes involved in the whole $\beta$-delayed decay scheme, namely, $\beta$-decay, proton emission and electromagnetic de-excitation. Our shell-model results successfully, though not fully, match with the key features of these experimental partial-decay schemes.