If a character has Improved Movement: Ignores Characters and is adjacent to an opposing character, can they still use the Move and Attack transporter ability?

Looking at Con Exclusive Superman & Wonder Woman as an example.

No, they can't.

Move and Attack
This character can use Hypersonic Speed if it is not adjacent to an
opposing character when given the power action, but its attack
value is modified by -2.

You can't be adjacent when the power action is given. You will not start ignoring character before you are moving and you can't be moving without given the action.