tl;dr version: Akrasia might be like an eating disorder!

When I was a teenager, I ate ice. Lots of ice. Cups and cups and cups of ice, constantly, all day long, when it was freely available. This went on for years, during which time I ignored the fact that others found it peculiar. ("Oh," I would joke to curious people at the school cafeteria, ignoring the opportunity to detect the strangeness of my behavior, "it's for my pet penguin.") I had my cache of excuses: it keeps my mouth occupied. It's so nice and cool in the summer. I don't drink enough water anyway, it keeps me hydrated. Yay, zero-calorie snack!

Then I turned seventeen and attempted to donate blood, and was basically told, when they did the finger-stick test, "Either this machine is broken or you should be in a dead faint." I got some more tests done, confirmed that extremely scary things were wrong with my blood, and started taking iron supplements. I stopped eating ice. I stopped having any interest in eating ice at all.

Pica is an impulse to eat things that are not actually food. Compared to some of the things that people with pica eat, I got off very easy: ice did not do me any harm on its own, and was merely a symptom. But here's the kicker: What I needed was iron. If I'd been consciously aware of that need, I'd have responded to it with the supplements far earlier, or with steak¹ and spinach and cereals fortified with 22 essential vitamins & minerals. Ice does not contain iron. And yet when what I needed was iron, what I wanted was ice.

What if akrasia is experiential pica? What if, when you want to play Tetris or watch TV or tat doilies instead of doing your Serious Business, that means that you aren't going to art museums enough, or that you should get some exercise, or that what your brain really craves is the chance to write a symphony?

The existence - indeed, prevalence - of pica is a perfect example of how the brain is very bad at communicating certain needs to the systems that can get those needs met. Even when the same mechanism - that of instilling the desire to eat something, in the case of pica - could be used to meet the need, the brain misfires². It didn't make me crave liver and shellfish and molasses, it made me crave water in frozen form. A substance which did nothing to help, and was very inconvenient to continually keep around and indulge in, and which made people look at me funny when I held up the line at the drink dispenser for ten minutes filling up half a dozen paper cups.

So why shouldn't I believe that, for lack of some non-food X, my brain just might force me to seek out unrelated non-food Y and make me think it was all my own brilliant idea? ("Yay, zero-calorie snack that hydrates, cools, and is free or mere pennies from fast food outlets when I have completely emptied the icemaker! I'm so clever!")

The trouble, if one hopes to take this hypothesis any farther, is that it's hard to tell what your experiential deficiencies might be³. The baseline needs for figure-skating and flan-tasting probably vary person-to-person a lot more than nutrient needs do. You can't stick your finger, put a drop of blood into a little machine that goes "beep", and see if it says that you spend too little time weeding peonies. I also have no way to solve the problem of being akratic about attempted substitutions for akrasia-related activities: even if you discovered for sure that by baking a batch of muffins once a week, you would lose the crippling desire to play video games constantly, nothing's stopping the desire to play video games from obstructing initial attempts at muffin-baking.

Possible next steps to explore the experiential pica idea and see how it pans out:

Study the habits of highly effective people. Do you know somebody who seems unplagued by akrasia? What does (s)he do during downtime? Maybe someone you're acquainted with has hit on a good diet of experience that we could try to emulate.
If you are severely plagued by akrasia, and there is some large class of experiences that you completely leave out of your life, attempt to find a way to incorporate something from that class. See if it helps. For instance, if you are practically never outdoors, take a short walk or just sit in the yard; if you practically never do anything for aesthetic reasons, find something pretty to look at or listen to; etc.
You might already have periods when you are less akratic than usual. Notice what experiences you have had around those times that could have contributed.

¹I was not a vegetarian until I had already been eating ice for a very long time. The switch can only have exacerbated the problem.

²Some pica sufferers do in fact eat things that contain the mineral they're deficient in, but not all.

³Another problem is that this theory only covers what might be called "enticing" akrasia, the positive desire to do non-work things. It has nothing to say about aversive akrasia, where you would do anything but what you metawant to do.