A few points.
Test strips are notoriously unreliable. You really can't trust what they tell you. They don't test for ammonia which is the most important test as that's the first parameter to go off if something is wrong, so you have to buy separate ammonia tests which makes them both uneconomic and unreliable.
However let's take your tests at face value.
Nitrate + nitrite. The presence of nitrate and no nitrite is an indicator your cycle is established, which is good, but nothing to do with your plant issues. You are correct that low nitrate is good for fish, but its bad for plants. I mentioned nitrogen in my above post, and plants typically get their nitrogen from nitrate. 10ppm is at the low end of what's a good parameter for plant health. Somewhere in the range of 10 to 20ppm nitrate is going to be a good level for both your plants and fish. If your test was after a water change that's probably going to be OK as the nitrate levels will increase before your next water change and keep you in the 10 to 20ppm range. If your test was before your water change it could do with being a bit higher, as your water change will lower the nitrate level below that 10ppm, and then gradually increase back up to 10ppm before your next water change again lowers it.
You do have a general hardness test on there. Your GH is below 6 degrees, which is considered soft. So your water doesn't contain much calcium in it. This could either be down to my point about the accuracy of test strips, or you are mistaken about your water being hard.
I agree your KH is somewhere in the 3 to 6 range, which again would be considered low. Low nutrient demand plants often get their carbon needs from KH, so plants usually do better in harder water.
So your water if we accept the test results as accurate isn't ideal for healthy plant growth.
You have a few options.
Nitrate. You could cut back on water changes to see if that gets the nitrate a little higher. But you arent really changing much water as it is, so I wouldn't reduce further. Add more fish maybe. Or, most aquarium fertiliser products don't have a lot of nitrogen (nitrate) in them as they are perceived as being responsible for algae. What fertiliser are you using? Some fertilisers do have nitrogen in there though.
Water hardness. I would get an accurate test and not rely on the strips as they contradict what your own understanding of what your water hardness is. If we did take your KH as accurate, while it's soft it's not so soft that it would cause any issues. A question though. Does your tap water go through a water softener?
Algae is a result in inbalance of nutrients, carbon and light. From what you have said carbon and nutrients are both low, so how about your light? Do you know the specs of your light? Does your aquarium get any natural daylight? 6 to 8 hours of aquarium lighting is usually sufficient for healthy plant growth while not driving too much algae growth. Contrary to common sense your light might be too much. If you have a strong light, or if its turned on for too long what will happen is that the light will drive plant growth, but as there are insufficient nutrients and carbon that plant growth can't be healthy. Too much light will also cause algae growth. Both of which you are seeing.
And a couple of final points.
Not all plants will do well in all aquariums. Keep the ones that do well and get more of those, and remove the ones that don't do well. I've tried numerous plants in various tanks to find the ones that do well in my particular set up.
Keeping plants is about trial and error. Try something, only one thing, and observe how it affects your plants over a few months. Any change, positive or negative, can then be put down to that one change. And then adjust something again and see what effect that has, again over a few months. So you might try reducing the light intensity, or duration. Or you might try a better plant fertiliser. If you change too many things at once you don't which change caused whatever effect you observe.