mowing will help a lot against your weeds
the ash or lime will reduce acidity and therefore give more plants an opportunity. egg shells or mussels or something like that will also work, but slower --> less agressive in the beginning, but also much more constant, what your grass will like more.
Or grow some acidity loving plants over there?
As you are 'complaining' about trees in the north, I suppose you are in southern hemisphere?
in other words, do the trees are in the sunpath or is the sun comming under the branches?
strawberries like it a little bit acid, but they also need sunlight!
ow, and I forgot, 'weed' does not exist
except the stuff some people smoke