adobe photoshop - How can I cut out certain parts of text?

This is the logo I have in mind:

I have been using white rectangles as blocs to hide the words like this:

However, when I merge the layers of text and white rectangles together, I am seeing something like this when I turn my background transparent:

It's very time consuming and laborous to manually delete those white spots in the final layer. If I change the color of the white rectangles to transparent then I am not seeing any "cutting" of the letters because it's transparent!!

Is there a better way to do this?