Let $f$ be a function of two variables $x$ and $y$. Assume that $f$ is $C^1$. Assume that $f_{xx}$ exists and continuous.
I suspect that the answers are negative, so let me ask a more general question.
Question If f is $C^k$ and $\partial f^n/\partial x^n$, $n>>k$, exists and continuous. Can one say anything about $f_x$ better than $C^{k-1}$?
From Counterexamples in Analysis (9.10), consider the function $f(x) = xy\frac{x^2-y^2}{x^2 +y^2}$. A bit of calculation confirms that $f_{xx}$ exists and is continuous, while $f_{xy}$ is discontinuous at the origin.
In general, consider, for example, $f(x,y) = \frac{x^k y^3}{x^2+y^2}$. You can take any number of $x$ derivatives, and the result will still be continuous, but if you take $k$ $x$-derivatives, and then a $y$-derivative, you will have a term of the form $\frac{y^2}{x^2+y^2}$ in your sum, which is problematic at the origin. So you can't get better than $C^k$ for this function.
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 $u_{xx}-u_y=g$.
