Let, $f:\mathbb{R}^{J+1}\to\mathbb{R}$, $g:\mathbb{R}^J\to\mathbb{R}$ and consider the following equation: $$f(x,g(x))=g(x).$$ I would like to know if the following statement is true (or find conditions under which this is true):
There exists $h:\mathbb{R}^2\to\mathbb{R}$ such that: $$f(x,y)=h(g(x),y).$$
In other words, neither $f$ or $g$ are known (although if necessary one could impose continuity properties), and hence I would like to know if the equation implies that $f$ has a representation in terms of $h$.
The assumption you have is that the restriction of $f$ to the graph of $g$ agrees with the projection onto last coordinate. This tells you pretty much nothing about the behavior of $f$ outside of the graph of $g$, even if you assume infinite smoothness of both functions. In particular, there is no basis for conclusion that $f$ can be expressed in terms of $g$ and the last coordinate.