Does precipitated copper carbonate contain impurities
Yes, the water contained in the freshly filtered precipitate will contain dissolved $\ce{Na2SO4}$, as the latter is the soluble by-product of your double displacement reaction. To get rid of it, you need to wash your precipitate thoroughly with distilled water (pH 7). If you really want to make sure that all $\ce{Na2SO4}$ has been removed, you can test samples of the wash water filtrate for $\ce{SO4^2-}$ by adding diluted hydrochloric acid and aqueous $\ce{BaCl2}$. When a white precipitate of $\ce{BaSO4}$ no longer forms, the aqueous sodium sulfate can be considered separated from the $\ce{CuCO3}$ precipitate.