Can't add gloss to logo on illustrator without making the background shiny

Hi Adobe Community

I am trying to add some shine to the lettering on my logo. I used the Rectangle Tool to create the shape of the shine and reduced its opacity to produce a glossy effect. However, as can be seen by the screenshot, there is excess shine [circled in red] affecting everything in the background. I would like to remove it from my logo, but I do not know how, except by imprecisely and painstakingly using the Eraser Tool. Any suggestions on how to remove it? Maybe by creating a mask? Also, a side note: the logo is a single layer comprising objects (not sublayers), and the shine is a separate layer.

Thanks and best regards

Alex