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We present a joint theoretical and experimental study of the oxygen $K$-edge spectra for LaFeO$_3$ and homovalent Ni-substituted LaFeO$_3$ (LaFe$_{0.75}$Ni$_{0.25}$O$_3$), using first-principles simulations based on density-functional theory with extended Hubbard functionals and x-ray absorption near edge structure (XANES) measurements. Ground-state and excited-state XANES calculations employ Hubbard on-site $U$ and inter-site $V$ parameters determined from first principles and the Lanczos recursive method to obtain absorption cross sections, which allows for a reliable description of XANES spectra in transition-metal compounds in a very broad energy range, with an accuracy comparable to that of hybrid functionals but at a substantially lower cost. We show that standard gradient-corrected exchange-correlation functionals fail in capturing accurately the electronic properties of both materials. In particular, for LaFe$_{0.75}$Ni$_{0.25}$O$_3$ they do not reproduce its semiconducting behaviour and provide a poor description of the pre-edge features at the O $K$ edge. The inclusion of Hubbard interactions leads to a drastic improvement, accounting for the semiconducting ground state of LaFe$_{0.75}$Ni$_{0.25}$O$_3$ and for a good agreement between calculated and measured XANES spectra. We show that the partial substitution of Fe for Ni affects the conduction-band bottom by generating a strongly hybridized O($2p$)-Ni($3d$) minority-spin empty electronic state. The present work, based on a consistent correction of self-interaction errors, outlines the crucial role of extended Hubbard functionals to describe the electronic structure of complex transition-metal oxides such as LaFeO$_3$ and LaFe$_{0.75}$Ni$_{0.25}$O$_3$ and paves the way to future studies on similar systems.