Hi Jonathan, glad you like the site and books.
On the whole I would pull back the covers and spread this manure under them, to feed soil life and give you a better soil for when your weeds are finally gone. I am unsure when you put the covers on, what weeds etc and that affects the answer, also what you plan to grow because spreading strawy manure can make it tricky to grow carrots and salads because of slugs and the difficulty of sowing small seeds. Best look on it a part of a "long term clean up" and grow squash family, brassicas, potatoes etc this year.