If you take an ordinary sized plastic water bottle full of water and pour a packet of powdered (or liquid) drink mix into it, Will shaking the bottle with the cap screwed on to dissolve the mix into the water work better if there is a small amount of air inside the bottle? Or does it not make a difference at all and will shaking work just as well with no air bubbles at all in the bottle?
Within reasonable limits, the more air in the bottle the better the mixing.
This isn't to do with air bubbles. To get good mixing you need turbulent flow, and for that you need high flow velocities. If the bottle is completely full it's hard to get a high flow velocity started because for water to move it has to push other water out of the way. If the bottle is only half full the moving water only has to push air out of the way, and air is both much more compressible and much less viscous than water. For any given amount of shaking effort you'll get higher flow velocities and therefore better mixing if there is a substantial amount of air in the bottle.
Compare the added liquid or powder with little balls which have about the same density as the water. If there is no air in the bottle you can shake it, but this will have little effect on the distribution of the little balls in the fluid. The balls, which without shaking will eventually distribute themselves uniformly in the liquid and will take a while, will make the same movement as the water and this doesn't reduce the time to distribute them throughout the water (it will take about the same time as just letting the bottle stand still and wait until the stuff is uniformly distributed throughout the water).
If there is an amount of air in the water and you shake the closed bottle the little balls are caught by the violently moving water and thereby get evenly distributed, which will also be the case if you add another liquid or powder. For the sake of simplicity, I wrote that the balls (or the added powder or liquid) have the same density as the water. If they have a lower or higher density the water and added stuff will separate into two layers after the shaking the bottle with air.