Do you think that improving communication skills is more important than improving customer focus skills, especially for new hires?

Yes, that right, communication skills to helps during conversation between new employee and boss. It need for certainty mastering the assigned new tasks and proper implementation.

I think that both are close to eachother and should be integrated , customer centericty or "customer focus" is to find out what is customer needs starting from the product design to the implementation and after sales as well. so preparing and to train our fromtliners to be good communcators is a vital elments in customer focus skills.

Both are very important just like the words! You can't only improve communication skills and skip customer focus skills. As I know, they work together!