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We study the geodesic deviation (GD) equation in a generalized version of the Sáez--Ballester (SB) theory in arbitrary dimensions. We first establish a general formalism and then restrict to particular cases, where (i) the matter-energy distribution is that of a perfect fluid, and (ii) the spacetime geometry is described by a vanishing Weyl tensor. Furthermore, we consider the spatially flat FLRW universe as the background geometry. Based on this setup, we compute the GD equation as well as the convergence condition associated with fundamental observers and past directed null vector fields. Moreover, we extend that framework and extract the corresponding geodesic deviation in the \emph{modified} Sáez--Ballester theory (MSBT), where the energy-momentum tensor and potential emerge strictly from the geometry of the extra dimensions. In order to examine our herein GD equations, we consider two novel cosmological models within the SB framework. Moreover, we discuss a few quintessential models and a suitable phantom dark energy scenario within the mentioned SB and MSBT frameworks. Noticing that our herein cosmological models can suitably include the present time of our Universe, we solve the GD equations analytically and/or numerically. By employing the correct energy conditions plus recent observational data, we consistently depict the behavior of the deviation vector $η(z)$ and the observer area distance $r_0(z)$ for our models. Concerning the Hubble constant problem, we specifically focus on the observational data reported by the Planck collaboration and the SH0ES collaboration to depict $η(z)$ and $r_0(z)$ for our herein phantom model. Subsequently, we contrast our results with those associated with the $Λ$CDM model.