= Solution
A generic linear projection over an infinite <field> is not enough for the stated arbitrary <field> $k$. Instead use <Noether normalization by weighted substitutions> to make the defining polynomial monic in one coordinate.
For $n>1$, choose an integer $M$ at least every exponent occurring in a nonzero <monomial> of $f$, put $B=M+1$, and use the triangular polynomial change of variables
$$
x_i=y_i+y_n^{B^i}\quad(1\leq i<n),\qquad x_n=y_n.
$$
Here $M\geq1$ because the irreducible polynomial is nonconstant. This is an automorphism, with inverse $y_i=x_i-x_n^{B^i}$, $y_n=x_n$. A <monomial> $x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ contributes a highest $y_n$ power
$$
w(\alpha)=\alpha_n+\sum_{i=1}^{n-1}\alpha_iB^i.
$$
These weights are all distinct: the exponents are base-$B$ digits bounded by $M$. All other terms in the expansion have smaller $y_n$ degree than that <monomial>'s weight. Therefore the highest-degree $y_n$ term in the transformed $f$ comes from exactly one original <monomial> and has a nonzero coefficient in $k$, independent of the other $y_i$. No assumption about the cardinality or characteristic of $k$ enters this argument.
Multiply by the inverse of that coefficient to obtain a <monic polynomial> $F\in A[y_n]$ of some degree $D>0$, where $A=k[y_1,\ldots,y_{n-1}]$. The same construction for $n=1$ simply rescales $f$ to be monic over $A=k$.
The <monic polynomial quotient is finite free> because division by $F$ gives a unique representative of degree less than $D$. Thus
$$
\boxed{k[x_1,\ldots,x_n]/(f)\cong A[y_n]/(F)
\cong A\oplus Ay_n\oplus\cdots\oplus Ay_n^{D-1}}
$$
as $A$-modules. Uniqueness follows because any nonzero multiple of a monic $F$ has $y_n$ degree at least $D$. In particular the map $A\to A[y_n]/(F)$ is injective.
The induced morphism
$$
\boxed{X\longrightarrow\operatorname{Spec}A=\mathbb A_k^{n-1}}
$$
is finite, since its coordinate algebra is a finite module, and flat, since tensoring with a <finite free module> is a finite <direct sum> of copies of the original module and preserves exactness. This is a <finite flat Noether normalization of an affine hypersurface>, with the explicit projection coordinates $y_i=x_i-x_n^{B^i}$. The construction actually applies to any nonconstant polynomial, not only an irreducible one.
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