In this paper we study the number of rational points on curves in an ensemble of abelian covers of the projective line: let normalℓ be a prime, q a prime power and consider the ensemble Hg,normalℓ of normalℓ ‐cyclic covers of Pdouble-struckFq1 of genus g. We assume that q ≢0,1modℓ. If 2g+2ℓ−2 ≢0mod(normalℓ−1)ordnormalℓ(q), then Hg,normalℓ is empty. Otherwise, the number of rational points on a random curve in Hg,normalℓ distributes as ∑i=1q+1Xi as g→∞, where X1,…,Xq+1 are independent and identically distributed random variables taking the values 0 and normalℓ with probabilities (ℓ−1)/ℓ and 1/ℓ, respectively. The novelty of our result is that it works in the absence of a primitive normalℓ th root of unity, the presence of which was crucial in previous studies.