You probably have overdispersion which is the most frequent reason for misfit of poisson models. Overdispersed poissons are fit with negative binomial or, if too much overdispersed, with poisson inverse gaussian. If your model excludes zeros you can use the zero truncated versions of these distributions.

The software that can do these models is gamlss package in R. Truncated distros are in gamlss.tr.

As for literature, I found Joseph Hilbe's books Negative Binomial Distribution and Modeling Count Data helpful for my models.

Thank you for your quick responses!!

Try Zero-truncated NB and some other Poisson mixed models.