Let $\mathcal{H}=\{\xi_1, \xi_2, ..., \xi_n\}$ be an alphabet - it may be an empty set - and $\mathcal{Y}=\{\alpha,\beta\}\cup\mathcal{H}$ and $\mathcal{Z}=\{\gamma_1,\gamma_2,...,\gamma_k\}\cup\mathcal{H}$ two other alphabet such as $\{\alpha,\beta,\gamma_1,\gamma_2,...,\gamma_n\}\nsubseteq\mathcal{H}$.
I have the following definition for the translation from $\mathcal{Z}$ to $\mathcal{A}$ $\textit{via}$ $\mathcal{H}$: $$\begin{align} \begin{cases} \xi_{i, \in \mathcal{Z}}\rightarrow\xi_{i, \in \mathcal{Y}}\\ \gamma_i \rightarrow \alpha\beta^i\alpha \end{cases}. \end{align}$$
For example, the translation of the english alphabet to $\mathcal{A}=\{\bullet,\bigcirc\}$ via the empty alphabet $\emptyset$ gives \begin{align} \begin{cases} a&\rightarrow\bullet\bigcirc\bullet\\ b&\rightarrow\bullet\bigcirc\bigcirc\bullet\\ c&\rightarrow\bullet\bigcirc\bigcirc\bigcirc\bullet\\ &\vdots\\ m&\rightarrow\bullet\bigcirc^{13}\bullet\\ &\vdots\\ z&\rightarrow\bullet\bigcirc^{26}\bullet \end{cases}, \end{align} and $$math\rightarrow \bullet\bigcirc^{13}\bullet\bullet\bigcirc\bullet\bullet\bigcirc^{20}\bullet\bullet\bigcirc^{8}\bullet$$.
Why do we need (or prefer) to have $\gamma_i \rightarrow \alpha\beta^i\alpha$ rather than $\gamma_i \rightarrow \alpha\beta^i$, as the translation is "lighter" the second way? Is it for an easier manipulation of the result?
That is not a standard definition or a standard concept. It looks like something a book or article might define as an ad-hoc stepping stone to something else.
Why it would choose to define it that way (rather than one of the myriad other possible ways to encode a larger alphabet into smaller) is impossible to answer without knowing what your source is actually doing using the concept it defines for.
Since the encoding (as well as your proposed alternative) is rather wasteful, one might speculate that its aim is primarily theoretical, and that the author has gone for what he/she thought would be quickest to explain, rather than trying to optimize for encoding efficiency.