If you have live plants in your tank, you will need some levels of nitrates since it is one of the main macro nutrients plant absorb along with phosphorus and potassium (NPK). Also, limiting nutrients for plants, can cause imbalances that can really give algae a foothold in your tank.
If you can manage to manually remove most of the hair algae, you can purchase some Seachem Excel (if you don't have it already) and squirt it on any remaining algae. This should kill it off, and then you can use the rest of the bottle as a carbon supplement for your plants. Hopefully, once your hornwort starts growing, it will out-compete the algae for nutrients, and you won't see so much of it.