This paper investigates the analysis of $\mathrm{b}$-generalized skew derivations, denoted as ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$, within a prime ring $\mathcal{R}$ with characteristic different from 2. Here, ${\mathcal{Q}}_r$ represents the right Martindale quotient ring of $\mathcal{R}$, and $\mathcal{C}$ denoted its extended centroid. Additionally, $\mathcal{L}$ is a noncentral Lie ideal of $\mathcal{R}$. Assuming ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ are nontrivial $\mathrm{b}$-generalized skew derivations associated with the same automorphism $\alpha$, the paper aims to explore the detailed structure of these generalized derivations that satisfy the specific equation:

$$pu{\mathrm{\Delta }}_1(u)+{\mathrm{\Delta }}_1(u)uq={\mathrm{\Delta }}_2(u^2),\text{with}p+q\notin \mathcal{C},\text{for all}u\in \mathcal{L}.$$

The above-studied result generalized the already existing results ^[1,2] in the literature.

In this paper, we investigate the structure of generalized derivations and quasiderivations of the tensor product algebra $\mathcal{H}\otimes \mathcal{A}$, where $\mathcal{H}$ is a Hom-Lie algebra and $\mathcal{A}$ is a commutative associative algebra with unity over the field $\mathcal{K}$. We examine whether every generalized derivation (respectively, quasiderivation) of $\mathcal{H}\otimes \mathcal{A}$ can be expressed as a sum of a derivation and a linear map in the centroid of $\mathcal{H}\otimes \mathcal{A}$, provided that this property holds for $\mathcal{H}$. We establish a partial affirmative result under suitable conditions and discuss the algebraic implications of our findings.