I can't speak to your specific clearing of throat and cough, but for many here that has been true in the early stages of their quits. The little cilia which have been paralyzed by smoking are coming back to life. Which is a GOOD thing. And so they're attempting to clear out all the cr*p that they haven't been able to before in our lungs.
What can relieve it? Perhaps some of the OTC mucus- busting medicines out there, certainly one can't drink enough water. So - drink water! Water helps to flush our systems in many ways. But I'd say just - "go with the flow" for a bit. It should calm down in not too long a time. Obviously if it keeps up for quite a while,, then it needs to be checked out.
Personally, I've squirted water up my nose (solution below) so it comes down the other nostril side - very unpleasant process, but it does help clear the mucus out. This is the solution my ENT doc recommends (almost demands! lol) from his office: "1 quart clean container filled with purified water, I think. Didn't write that down Maybe distilled, but as I recall it was purified. . Add 2-3 heaping tsp canning salt + 1 teasp baking soda. Squirt up nose twice a day." Or you could try a neti pot. I just went with the direction from my doc. Obviously the salt is to dry up the mucus and the baking soda is to remove bacteria. At least that's what I assume. I wasn't told what it did to help, just that I should employ that technique. I am NOT a doctor and I am only giving you my own doctor's recommendation. Below is an image of the type of bulb I used.
Again, it's not a very pleasant process. And definitely hold your face over the sink if you should try it. But I found it did help to clear my sinuses - kind of.
Here's a link to another doc recommending the same thing: https://www.bellin.org/images/uploads/NORMAL_SALINE_IRRIGATION_RECIPE_AND_INSTRUCTIONS.pdf
