Let $C$ be a cubic curve defined over a field $k$. Take, for example, an affine curve:
$$ C = \{(x,y) \in k\times k : a x^{3} + b x^{2} y + c x y^{2} + d y^{3} + e x^{2} + f x y + g y^{2} + h x + i y + j = 0 \} $$
I know that if the curve were singular, then it would only have one singular point with coordinates in a fixed algebraic closure $\bar{k}$ of $k$. My question is the following: in which cases will this point be a $k$-rational point of the curve.
If $k$ is a perfect field, I (think I) know the answer: assuming one of the coordinates of the singular point, say, $x$, is not in $k$, its minimal polynomial $m(x,k)$ will have more than one root and, through an automorphism of $\bar{k}$ taking $x$ to another root of $m(x,k)$, one will end up with more than one singular point. However, in the case in which the field $k$ is not perfect, I cannot find an answer.