Autonomous or non-autonomous optimal control system?
What about rewriting the problem in the following way?

$$\underset{\left\{ c_{t}\right\} }{max}\int_{t=0}^{\infty}\left[u\left(c_{t}\right) e^{y_t}\right]e^{-\rho t}dt$$

with the new state variables defined as

$$\begin{align}
\dot{k_{t}}=f\left(k_{t}\right)-c_{t}\\
\dot{y}_{t}=-h\left(k_{t}\right)
\end{align}$$

given the initial conditions $(k(0),y(0))=(k_0,0)$.