If you're talking about convergence under the $L2$-norm, I think the answer is probably no.
For example, take $M_t = 1$, and $g(t)$ as a step function. The "derivative" of $g(t)$ is a sum of delta functions, which are not in $L2$ (??).
However, the "derivative" of the function may exist in the following sense.
For a given interval $[a,b]$, consider two cases: 1. $M_t$ is flat (i.e. no jump occurs in $[a,b]$) 2. At least a jump occurs in $[a,b]$.
For the first case, it can be easily shown that $h(t) - h(a) = \int_a^t M_t f(t)^{M_t-1} d f(t)$. The integral exists in the "classical" sense because $f(t)$ is increasing and right-continuous. (not sure; please verify)
For the second case, even if $f(t)$ is constant, $f(t)^{M_t}$ will be discontinuous where $M_t$ "jumps". The accumulative size of "jumps" up to $t$ is given by $\int_a^t f(t)^{M_t} dM_t / M_t$. Again the latter integral seems to exist because $M_t$ is increasing and right-continuous. (Possible problem where $M_t = 0$)
Thus it seems that, $h(t) - h(a) = \int_a^t M_t f(t)^{M_t-1} d f(t) + \int_a^t f(t)^{M_t} dM_t/M_t$, the convergence being almost surely (or almost everywhere). (The latter formula may be very incorrect; please verify)
The idea is to obtain the set of points where either $f(t)$ or $M_t$ jumps. Of course it may not work mathematically.