Terahertz (THz) frequency bands can be promising for data transmissions between the core network and access points (AP) for next-generation wireless systems. In this paper, we analyze the performance of a dual-hop THz-RF wireless system where an AP facilitates data transmission between a core network and user equipment (UE). We consider a generalized model for the end-to-end channel with an independent and not identically distributed (i.ni.d.) fading model for THz and RF links using the $\alpha$-$\mu$ distribution, the THz link with pointing errors, and asymmetrical relay position. We derive a closed-form expression of the cumulative distribution function (CDF) of the end-to-end signal-to-noise ratio (SNR) for the THz-RF link, which is valid for continuous values of $\mu$ for a generalized performance analysis over THz fading channels. Using the derived CDF, we analyze the performance of the THz-RF relayed system using decode-and-forward (DF) protocol by deriving analytical expressions of diversity order, moments of SNR, ergodic capacity, and average BER in terms of system parameters. We also analyze the considered system with an i.i.d. model and develop simplified performance to provide insight on the system behavior analytically under various practically relevant scenarios. Simulation and numerical analysis show a significant effect of fading parameters of the THz link and a nominal effect of normalized beam-width on the performance of the relay-assisted THz-RF system.