Even if $f_{xx}$ and $f_{yy}$ both exists and continuous, it is possible that $f$ is not in $C^2$. This is a basic property of the solutions of $\Delta u = g$. One recovers $C^2$ if the right hand side is a bit more than merely continuous. In order to recover the full regularity gain of 2, one has to use (at least) Hölder or Sobolev spaces. The situation you have can be more directly handled by regularity theory of the heat equation $u_{xx}-u_y=g$.

This does not directly answer your question, but is a related phenomenon:

Even if $f_{xx}+f_{yy}$ exists and continuous, it is possible that $f$ is not in $C^2$. This is a basic property of the solutions of $\Delta u = g$. One recovers $C^2$ if the right hand side is a bit more than merely continuous (e.g. Dini continuity is enough). In order to get the full regularity gain of 2, one has to use (at least) Hölder or Sobolev spaces. The situation is similar for the heat equation $u_{xx}-u_y=g$.