How do you remove the hard edge from a gradient layer mask?

[defaultmj6u4pwzs69i]

I have a rectangular JPG graphic placed on top of a solid color background. I want to the graphic to fade to transparent on the right edge, blending into the background.

I cannot figure out for the life of me how to make it fully fade—there is always a harsh edge. Is there a way to achieve a gradient that fades softly all the way to transparent? I took a screenshot of just the problem area to demonstrate the issue I'm facing.

I am a generalist at a nonprofit trying to make it all work, so my apologies if this question reveals general ignorance of Photoshop's functionality. I feel like there should be a simple solution, and I'm losing my marbles after scouring the internet for answers for hours and trying everything I can think of. 

[Derek Cross]

Try painting with the edge with the Dodge tool..

[Stephen_A_Marsh]

Have you tried adjustments to gradient opacity sliders (midpoint or either end)?

[davescm]

Make sure the black area of your gradient is fully black (RGB0,0,0) and at full opacity where it crosses the hard edge of the image. Anything above that value will allow the edge to show through.

Dave

[Semaphoric]

When you drag a Gradient into the mask, place the start point of the gradient just inside the edge of the image, and drag the end into the image, holding the Shift key to constrain the direction.

Once done, when the mask and the Gradient tool are both selected, you will see the controls on screen, so you can adjust the position later, if desired.

[Trevor.Dennis]

Dave's explained what is happening with your mask and how to fix it, but another wee tip that might be useful is to unlink the mask by clicking on its chain icon.  You can then move the mask independently of the layer, and nudge it to the left to hide the hard edge.