normally I would recommend a bleach and water solution but this might not give you results you want.
This will date me, but when I was a Boy Scout, all we had were thick old canvas material for our tents. If we were camping and the last day was raining, we HAD to pack up our tents but we knew as soon as we got home to re pitch it and air it out and let the sun hit it.
On your situation, I would spread it out, let direct sunlight hit it and lightly rub white vinegar and water on the bad spots. Let sunlight and ventilation do the job. Heavy chemicals will cause fading and deterioration of the material.
When its completely dry and cleaned, KEEP a camp dry or scotch guard on the material all the time. This will cause the water to bead up and not penetrate into the weavings.