get the clay bar and give yourself a good day of detailing on the car.
The clay bar will remove git and dirt without removing paint leaving the surface properly cleaned, meaning when you polish it will go on and come off more easily, will give a better result, and the most important, wont be draging grit around as you polish.
t-cut should only be used to remove stuborn marks that wont shift with the clay as it works by cutting through the surface..
Clay is very quick and easy to use and will make future polishing sessions far less graft, do a quick search on youtube for it to see how it is done. It is 100 x easier than t-cuting a car.