If the work you are doing is not stimulating and is merely a means to get a salary, then of course, you will eventually be drained of motivation and energy and it gets harder to wake up for it with every passing day.

On the other hand, something to look forward to, might give you the spark to "rise and shine".

If on top of uninteresting work, you pick up an uninteresting side-project, you are bound to make things worse. However, if the side-project is something for which you are itching to make time, then you might in fact wake up with excitement for it.

Is there anything you enjoy doing that you could eventually be paid for? I would say that spending that morning time developing the skill that is fun for you could be a good use of your time.