Here's an alternative way to think about it, which doesn't use the powerful methods of the tame fundamental group:

You can easily deduce from Riemann-Hurwitz that the genus of X is 0, i.e. it is just the projective line. Look at the affine patch: t is not infinity. Above t=infinity there's only one point. So take that point out, and call the parameter of the resulting affine line h. Then we have the inclusion of k[t] in k[h]. So t=g(h) where g is a polynomial. Since over t=0 there is one point with ramification n, g has multiplicity n: g=u(h-c)ⁿ. So after change of variables g=u*hⁿ, as required.

Here's an alternative way to think about it:

You can easily deduce from Riemann-Hurwitz that the genus of X is 0, i.e. it is just the projective line. Look at the affine patch: t is not infinity. Above t=infinity there's only one point. So take that point out, and call the parameter of the resulting affine line h. Then we have the inclusion of k[t] in k[h]. So t=g(h) where g is a polynomial. Since over t=0 there is one point with ramification n, g has multiplicity n: g=u(h-c)ⁿ. So after change of variables g=u*hⁿ, as required.