Can bounding boxes further improve the performance of a CNN classifier? Suppose I have a standard image classification problem (i.e. CNN is shown a single image and predicts a single classification for it). If I were to use bounding boxes to surround the target image (i.e. convert this into an object detection problem), would this increase classification accuracy purely through the use of the bounding box? 

I'm curious if the neural network can be "assisted" by us when we show it bounding boxes as opposed to just showing it the entire image and letting it figure it all out by itself.
Another way to ask the question is: Does sound get clearer when you remove the background noise? 

The obvious answer is yes and in the case of image classification, the answer is also generally yes.

In most cases reducing the noise (irrelevant pixels) will strengthen the signal (activations) the neural network is trying to find.