Can you use exploit weakness with HHS, Charge Transporter or any other move and attack ability.

It depends. You cannot give more than one non-free action to a character each turn. And one action can only activate one power that requires you to give the character that kind of action to be activated.
BUT some powers grants the character free actions during their resolution, and if said free actions are of the right kind, you can use them to activate a power if you like.
Now, let see the details of your question :
Exploit weakness requires you to give the character a close-combat action to activate, so it won't be usable during another action unless you are using an effect that specifically say otherwise.
HSS and the Move and Attack ability (that allows you to use HSS with some changes) grants the character the ability to make a close-combat (or ranged combat) attack (not action) as a free action. This is NOT a close combat action, so you won't be able to activate EW during HSS.
Charge, on the other hand, specifically state that once the move part of the action is over, the character can make a close-combat action as a free action. It is an action, so it can be used to activate any power requiring a close-combat action to activate. EW requires that type of action, so it's fine to use EW with Charge.
As for other powers or abilities that allows a character to move and attack, it depends on the text of said effect : if it allows the character to make a close-combat action (like the Green SCar SP), then it will work just fine ; if it only allows the character to make a close-combat attack (like The Thing SP) then it won't work.
I hope it helps.

Rep to you sir for giving a detailed and well explained answer. Too often I see people simply answer Yes or No to these questions, without an explanation as to why. Understanding the rules of the game is far more important than just knowing yes or no.

This is correct.

Thanks for the compliment, really

Thank you for the perfect answer and explanation. Alot of people assume it just works with everything. I tried to explain the same way you did but I wanted a little verification.

and exploit weakness works with steal energy (cause all close combat action are also close combat attacks) and B/C/F (cause EW is a close combat action), flurry works BCF too (though not sure if you roll for each attack or roll before and your attacks are that...) but flurry doesn't work with EW (because they are their own actions)

BCF is rolled after a successful attack roll. So if you use Flurry and BCF, you will roll for BCF after each successful attack. (So twice hopefully.)