I suggest a logistic regression model, defined over individual judgements rather than the aggregate score. Response variable is whether the vote was good or not. Predictors are product id and category.
The test is a standard test for significance of a predictor in a logistic regression model for the category factor. For example, compare the two models using a chi squared test (full model vs model without category factor).
Edit: if @Scortchi is correct and each product only occurs in one category (which makes sense), then his suggestion is correct.
The reason you want to include a predictor for product is that different products are presumably of different quality and so there's variation between products irrespective of which category they're assigned to. If you don't capture this in your test you're missing part of the information and your test will have less power.
I suggest using a regression model because many tests on more complex data with multiple factors can be viewed as tests on regression models. If you do as I say and treat each judgement as a data point instead of each product, you'll naturally incorporate the frequency bias you want (more popular product = more data points) in a principled way, instead of testing against a derived and I would argue flawed aggregate metric.