Construction of a branched cover of a rational curve For the affine line $\mathbb{A}^1_k:=\text{Spec}\,k[t]$, the $k$-algebra homomorphism induced by $t\rightarrow t^n$ gives a covering morphism $\mathbb{A}^1_{k} \rightarrow \mathbb{A}^1_{k}$ branched over $0$, where $(\text{char}\,k, n)=1$. Suppose now $C$ is a smooth affine algebraic curve over $k$, and $pt$ is a smooth $k$-valued point of $C$. My question is about this kind of cover in general, For every $n$ that is coprime with $\text{char}\,k$, does there exist an $n$-cover of $C$ that is only branched at $pt$ with multiplicity $n$ (like the affine line case). Replace $C$ by an open subset which contains $pt$ if necessary.

To construct a cyclic cover of degree $n$ over a scheme $X$ you need a line bundle $L$ with a (non-zero) morphism $f : L^n \to O_X$. Indeed, the covering can be defined as $$ Y = Spec_X(O_X \oplus L \oplus \dots \oplus L^{n-1}) $$ with the ring structure induced by the map $f$. The branch divisor of the covering $Y \to X$ is given by the equation $f = 0$.

Applying this construction in case $X = C$, with $L$ being the ideal of a point $P \in C$, and $f$ being the section of $L^{-n}$ corresponding to the divisor $nP$, one obtains a covering $Y \to C$ that is branched only over $P$.