Approximate a Poisson random variable as a functional of its intensity in a loglikelihood function

Question

I have an expression of a log-likelihood of $\theta=(c_1,c_2,\dots,c_J)$ which looks like \begin{equation} \sum_{j=1}^J \left( -c_j \sum_{m=0}^{M-1} g_j(x_{u_{m}}) \Delta u_{m} + \log(c_j) R_{j,[0,T]} + \sum_{m=0}^{M-1} P_{j}^{m} \log(g_{j}(x_{u_{m}})) \right) , \end{equation} where $P_{j}^{m}=Poisson(c_j g_{j}(x_{u_{m}}) \Delta u_m)$ and $\Delta u_m=u_{m+1}-u_{m} $. I want to write the last term as functional of $c_j$ so I can maximize the likelihood with respect to $\theta$ but I get stuck. Could anyone help? Thank you.