Simulating Noise in Astronomical Images
Yes, one can superpose these two noise sources: add a Gaussian readout noise (with a spatially constant sigma) to a Poisson noise (with spatially variable number count k).

In practice, as the number counts are high for optical CCD images, one often approximates the Poisson noise by a spatially-variable Gaussian noise, using the square root of the photon number counts as sigma. If your images have some significant sky background level (e.g., hundreds of photons), this is fine even for faint sources that add only few photons.  A Poisson distribution gets very close to a Gaussian already for k > 20. Note that despite this common approximation by a Gaussian, such a simulated noise contribution is still often called "Poisson noise", to stress that it originates from the spatially variable signal.

One warning: using the square root of the signal as sigma of a Gaussian to approximate Poisson noise only holds for a signal in units of photons (or almost equivalently, in units of electrons). If an image is given in ADU (analog to digital unit), the gain is required to convert the pixel values into electrons. The same holds when using a true Poisson distribution: one has to be careful that the counts have not been rescaled.