The idea of a "focus mode" or "distraction free mode" is exactly the wrong framing for writing. You don't want to focus on the nothing and emptiness of a page or a screen. You want to start by focusing on an idea and preferably many ideas. Do this first and then proceed from there.

Given the diversions of modern digital life, perhaps the best way to do one's writing is to do it at the moment of reading the actual references. Often while reading, one isn't as apt to have their attention diverted by the vagaries of life, instead they are focused on the thing at hand. It is while one has this focused attention that they should let their note taking practice while reading take over.

Even if you are distracted, you can at least maintain focus on a single line of text and your thoughts related to it and write them down in either a summary sentence or with a few related ideas which are sparked by the initial idea.

(This note is such an example.)

Then one can start and complete a small idea at a time and then letting them build over time and space, then recollect them to create a piece which then doesn't need to be written and painfully created, but which may only need an outline structure and some final polish and editing.