The problem is that you're trying to go from apples to oranges - the gradient in Corel Draw / Photoshop is a raster image, and you're trying to then bring it in and apply it as a vector gradient. A raster gradient has a defined size, and attempting to map it using the live trace tool causes Illustrator to try and find specific values at specific points within the gradient - thus the "stepped" result. If you're greatly expanding the raster shape you'll also be dealing with jaggies and aliasing artifacts from rescaling a raster gradient.
I'd recommend trying a couple of things:
- The gradient tool (shift + g) in Illustrator will allow you to fill a shape using the same types of controls as you'd see in Photoshop when building a gradient. I've used this very successfully as it's a lot simpler than the mesh tool (which is meant to provide precise gradient control over an irregular shape, not a nice gradual gradient). To use the regular gradient tool, create a shape and click on the little gradient under the foreground / background color swatches to fill the shape with the existing gradient.

The existing gradient can then be manipulated in the Gradient flyout menu.
- If that doesn't tickle your fancy you could always live trace the imported gradient and then use the mesh point removal tool (select the gradient tool and hold down option while clicking on a point) to delete most of the points in the mesh. It would still be a mesh but it would be a lot simpler and thus easier to deal with.