Question

Can a body have energy without having momentum

Answer 1

Yes, a body can have energy without possessing momentum. A body may have potential energy and still can be at rest. This means it would have zero momentum and still it could possess energy.
Eg: A brick placed on the top of a building has a PE = mgh, but it has got no momentum because it is not moving. ( Not moving => velocity is zero => mv = 0 ie Momentum is zero )

Hope this helps :)