In this paper, we further extend the study of function-correcting codes in the homogeneous metric over a chain ring $\mathbb{Z}_{2^s}$ for broader classes of functions, namely, locally bounded functions and linear functions, and for weight functions, modular sum functions. e define locally bounded functions in the homogeneous metric over $\mathbb{Z}_{2^s}^k$ and investigate the locality of weight functions. We derive a Plotkin-like bound for irregular homogeneous distance code over $\mathbb{Z}_4$, which improves the existing bound. Using locality properties of functions, we establish upper and lower bounds on the optimal redundancy. We provide several explicit constructions of function-correcting codes for locally bounded functions, weight functions, and weight distribution functions. Using these constructions, we further discuss the tightness of the derived bound. We explicitly derive a Plotkin-like bound for linear function-correcting codes that reduces to the classical Plotkin bound when the linear function is bijective, we further discuss a construction of function-correcting linear codes over $\mathbb{Z}_{2^s}$.