Do you mean Super Training?
If so, no. You will not gain double HP EVs holding a Power Weight and completing an HP Training mission.
The Power Items are for battling. However, it does stack with PokeRus, meaning that holding a Power Anklet (speed) and defeating a Magikarp (normally +1 Speed EV) will quadruple the EVs gained, gaining 4 Speed EVs.
Source: Experience