When does this nonlinear system of equations have a solution?

Question

Consider the following system of equations $$\begin{cases} ||x-k||_p = t\\ f(k+x) = f(k) \end{cases}$$ where $x \in R^n$, $k \in R^n$, $f: R^n \mapsto R$ is a convex function and $|| \cdot ||_p$ is the $L^p$ norm. Given $k$, $t$ and $p$, my aim is to either prove that the system has no solutions or that it has at least one solution (the actual solutions are not relevant). Can I apply Newton's method? Does imposing additional constraints on $f$ make the problem tractable?