It takes a huge amount of plants to have an impact on nitrate level....you'd need many fast growing stem plants, so many you'd barely see the fish. What can help is floating plants like salvinia which does suck up a fair bit of nitrate if you have plenty of it. But as your tap level is low, you really need to figure what it is that's raising the level so much in the tank. Have you tested the tap water recently? The level can vary in some areas, especially agricultural areas where fertiliser gets washed into the water courses from time to time.
Sorry I wasn't clear in my previous post. You need to remove the existing cyanobacteria and then black out completely for three days by covering the tank with a heavy blanket or black bin bags etc., so there is no light at all.