Variables that you want to control for are usually added as independent variables to your model (so you'd simply add price to the RHS of the equal sign in your model statement).
That being said, I wonder if you are approaching your modeling in an appropriate way? What is $p_1$ here? Is $p_1$ a count variable? Poisson models are usually reserved for modeling count data (i.e. when your response is a count, not your independent variables) -- this is the reason you were probably told to use a Poisson model -- not because of the first reason you listed in your post: "I have same person buying multiple products (also, a given product might be bought by several people)." When you have multiple purchases by the same person, this usually indicates that here is a correlated data problem, which means you'll likely have to rely on other methods than a standard Poisson model. This is likely why someone recommended you use a mixed model. That being said, given that you are interested in overall/population gender effects, and you have correlated data, I'd recommend the use of Generalized Estimating Equations or a Quasi-Poisson model. Mixed models are generally needed for estimating individual level effects (e.g. if you were interested in looking at a specific person's purchase behavior), but in this case, you seem interested only in population-level estimates to predict general preferences among demographic blocks.
What I would recommend you do is build a GEE model and include as many different attributes as you want to control for on the product as covariates in your model as well as your repeated subject ID and gender indicator variable. Then you'll be able to say for example what the expected effect on your response is when you go from males to femals (or vice-versa) when controlling for the given attributes. You should consider including interaction terms if you think certain attributes have a differential effect on your outcome variable for males and females. This will also allow you to assess the impact of the attributes on preferences.