Poisson model on moving averages: non-discrete

I have a model that is based on counts of cases. It follows a poisson distribution quite well but we are unhappy about the results of the model. In order to improve it we wanted to use moving averages of the counts instead of the counts themselves as the dependent variable, but this changes my variable into a non-discrete variable.

I can still run the analysis in R, but I get a warning message regarding the non-discrete nature of the variable.

Is this allowed? What are the things to look out for when using a poisson model with a non-discrete response variable?

Many thanks!
One solution is to use the moving total instead of the moving average. If you are using a simple moving average then this will just be a multiple of the average. This also shows that there really should not be a problem using the average itself. For many purposes, there are better smoothers than the simple moving average (e.g. Loess) but they are more complex