Poisson distribution with a constant term I have a set of data constituted by two columns: $Y$ with non-negative discrete values, depending on a time $t$.
I make the hypothesis that $Y$ is a Poisson distribution, with:
$$\operatorname {E} (Y\mid t )=e^{At + B} + C $$
When $C=0$ this is a well known generalised linear model, but how can find a good fit for data with an unknown value of $C$ ?

If you want to model your observations with $Y\lvert t \sim \text{Pois}\left(\lambda\left(t\right)\right)$ with $\lambda\left(t\right)=e^{At+B}+C$, then you will have:
\begin{align}
\mathbb{P}\left(Y=k\lvert t\right)&=e^{-\lambda(t)}\frac{\lambda(t)^k}{k!}\\
&=e^{-\left(e^{At+B}+C\right)}\frac{\left(e^{At+B}+C\right)^k}{k!}
\end{align}
If you have a sample of $n$ independent values $\{t_i, y_i\}, i=1,...,n$, you can use the equation above to define the log-likelihood of your sample as a function of your parameters $A,B,C$. Then, you can use an optimiser in order to find the parameters maximising the log-likelihood function. I would suggest using a parameter $C\prime=\ln(C)$ instead of $C$ directly in order to make sure that $C$ will be positive, in order to get $\lambda(t)>0$ $\forall t>0$.
