Let $A$ be a DVR of characteristic $p$, with $\pi$ a uniformising parameter, with $K=frac(A)$ the field of fractions. Consider the extension $L=K(\alpha)$ where $\alpha$ has minimal polynomial $y^p+\pi^b y+\pi^c$, with $0<b\leq c$. The problem is to show the existence of an element of $L$ with minimal polynomial $y^p+u\pi y+v\pi^{p(c-b)}$, with $u,v\in A^*$.
I think this is going to be manipulating the known minimal polynomial of $\alpha$ to produce such another element, eg $\alpha \pi^k+\pi^l$, for $k,l\in\mathbb{Z}$, possibly using the Frobenius map.
I wasn't able to make this work however, so it may require a more sophisticated idea. Any hints would be much appreciated.