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Let $l\ge 2$ be a positive integer, ${𝔽}_q$ a finite field of cardinality $q$ with $q\equiv 1(modl)$. In this paper, inspired by [, , ] and using a slightly different method, we study the fluctuations in the number of ${𝔽}_q$-points on the curve ${ℂ}_F$ given by the affine model ${ℂ}_F:Y^l=F\left(X\right)$, where $F$ is drawn at random uniformly from the set of all monic $l$-th power-free polynomials $F\in {𝔽}_q\left[X\right]$ of degree $d$ as $d\to \infty$. The method also enables us to study the fluctuations in the number of ${𝔽}_q$-points on the same family of curves arising from the set of monic irreducible...