Does writing matter a lot in research?
Writing matters a lot, and a lot of the time probably most PhDs (and even postdocs) face a steep learning curve. This curve is particularly bad in a field where writing is your tool, such as Philosophy. Even when analysis takes the form of symbols, in mathematics and logic, it is essential to convey your ideas clearly. Clarity is particularly problematic for young researchers, because of our tendency to hide behind pretension. Pretension, confusion, clunkiness must be removed for your ideas to come through.

Overcoming these issues is not essential to get an inexperienced researcher to approve of your work, but any researcher worth their salt will know when what you say is bullshit, or when what you say might have value but is unclear. I say this as an experienced, but not necessarily a good, writer. My writing has often been criticised and quite rightly too. I wish I had learned to write better earlier, is my point and the advice I give.

Because writing is important, there is a second question. How to focus on improving it?


Read good writing.  
Read a book on good writing, I recommend Joshua Schimel's Writing Science. 
Once you have written, wait, and then go back and read it. And then edit it. You will read it from the perspective of another person and hence realise why writing is so important and partly what you need to do to improve.