Showerhead with "backflow prevention"
What the backflow preventer does is keep the water that is in the head when you turn the water pressure off from flowing back into the supply pipes. This is actually a good thing, and it's required on all fittings that have a hose ... if not all fittings at this point, so that soap (or dirty water) doesn't leak back into your house's fresh supply lines and potentially contaminate your food. You really don't want to remove the backflow valve or alter the hose in any way.  

If it continues to leak out of the hose for hours afterwards, I would look at your valve assembly (specifically the diverter) as the culprit. You might have a small leak in the valve assembly that's flowing into your diverter and then up to the shower head. Does the tub filler operate correctly? What about taking the shower head/hose/etc. assembly completely off? Does water continue to leak out after you've turned the water off? 

Once you verify that the diverter and valve are shutting off correctly, you want to make sure that there are no cracks in the hose or fittings. I generally need to replace these shower heads once every 3-4 years. 1 year would be fast for one to go bad, but you never know with manufacturing of mass-produced plastic products. Last but not least, did you use the teflon tape just on the metal-to-plastic union, or did you also use it on the plastic threads near where you circled? I know it says not to, but I found that there's something to be said for a little extra teflon here and there.