Is this an instance of a Diffie-Hellman problem?
As stated the problem is insoluble as zero-knowledge of $k$ is provided (unless we know something about $m$).
To see this, let $x$ be any integer $0\le x\le p-1$ and let $m':= m\cdot v^{-x}$ then we see that $c=m'\cdot g^{(k+x)a}$ and we see that $k+x\pmod p$ is an equally valid output unless we have some reason to choose $m$ over $m'$.