Using a stand mixer with the paddle attachment or a hand mixer, combine softened cream cheese and sugar, mixing until incorporated, about 1 minute.
Add the eggs, one at a time, mixing about 30 seconds each time. Add the vanilla and mix until incorporated. Add the flour and mix for 20 seconds.
Pour the cheesecake filling over the cooled baked blondie and smooth the top.
Dollop the raspberry puree over the top of the filling and use a knife to swirl it into the filling.
Bake the cheesecake bars at 350 degrees for 36-40 minutes until cheesecake is almost set in the middle but still slightly jiggles and the edges are set and barely begining to turn brown.
Cool on a wire rack for 30 minutes and then place in the refrigerator to chill for at least 2 to 3 hours before slicing and serving.