Please see the GIFs here for a demonstration of convolutions:
https://github.com/vdumoulin/conv_arithmetic#convolution-animations
As you can see, convolutions operate the same regardless of the position in the image, so weight initialization cannot change the focus of the image.
It is also not advisable to rush into thinking about what the net will and won't need to learn your task. There are sometimes surprising amounts of signal outside what you as a human might focus on. I would suggest training the net and seeing how it performs, and then (as others have suggested) thinking about cropping.