Convoluted definition of a set: H(S)={x∈G∣∃n∈N,∃{x1,x2,⋯,xn}⊆S∪S−1,x=x1⋯xn}.

No. H(S) is the set of all elements g of G such that g can be expressed as a finite product of elements of ${\displaystyle S\cup S^{-1}}$.
Maybe lets have a look at example. Consider ${\displaystyle G=\{0,1,2,3\}}$ with addition modulo 4 as group operation, i.e. ${\displaystyle G={\mathbb{Z}}_4}$. Now let ${\displaystyle S=\left\{2\right\}}$. Since we are dealing with addition, then ${\displaystyle S^{-1}=\left\{2\right\}\text{because}2+2=0}$ in our group. Therefore ${\displaystyle S\cup S^{-1}=\left\{2\right\}}$.
Next H(S) is the set of all elements g of G such that ${\displaystyle g=x_1+\cdots +x_n}$ for some ${\displaystyle x_1,\cdots ,x_n\in S\cup S^{-1}}$. Since our last set has exactly one element then only possibilities for g are: ${\displaystyle 2,2+2,2+2+2}$, and which we can write in a compact way as ${\displaystyle n\cdot 2}$ for some ${\displaystyle n\in \mathbb{N}}$. By removing duplicates (remember that ${\displaystyle 2+2=0}$) this gives us
${\displaystyle H\left(S\right)=\{0,2\}}$