As is normal you are lumping a bunch of stuff all into the concept of "cycling".

There is bacteria that convert ammonia and nitrite. That's one kind of "cycling". Generally this is called bio filter cycling, and a bio filter can be just the sides of your tank, it doesn't have to be a separate filter. Algae is consuming all ammonia directly, so no ammonia for bacteria, so you'll not be able to "cycle" a bio filter.

When water goes from green to clear some people call that cycling. What natural process causes this is unknown. But of the theories a 50 gal stock tank wouldn't be a good candidate. That would leave two choices imo. Trickle water changes, but I think 50 gal tank would be much too small for that to be practical. That leaves a UV filter. 100% effect in 5-7 days.

I'm not a fan of terms like "cycling", "balanced", etc. They're used when there's little actual understanding. So I'd stay away from web sites and people who use such terms in a broad way.

It's so helpful when someone states things plainly. I could read for ages and not come across this in a way I could comprehend.

I had thought ammonia was a bad thing in a tank.

Is the algae related to the poor health of the hornwort and water lettuce?

New ponds don't have a lot of nutrients in them. The plants are likely reflecting that.

Algae doesn't hurt other, more desirable plants.

Ammonia is a very bad thing for fish but good thing for algae and bacteria that eat it. In the context of "cycling" a bio filter ammonia is needed to grow the bacteria. The bacteria require ammonia to reproduce and stay alive. You can't "cycle" a bio filter without ammonia.

If you used a UV to kill the green water then ammonia from fish would be available to bacteria, which would reproduce to a level where they were consuming all available ammonia. At that point the bio filter would be called "cycled". And again, bio filter can just be the sides of the pond.

Currently you could say the green water is your "bio filter" and that it is cycled because it, I assume, is consuming all available ammonia. When a pond clears that bio filter is gone as well.

Cycling is a very general term that really has little meaning by itself. Certainly confusing when used by itself.

If the hornwort you're talking about is the underwater kind then green water algae can kill it by settling on it, blocking light and taking up all available nitrate. Water lettuce can have a difficult time getting any nitrates as well because the green water algae are always first in the buffet line. People often say higher plants starve green water algae but the opposite it true. Lack of nitrate doesn't kill plants, they just can't grow without the certain nutrients.

Your goldfish can also nibble the roots which doesn't help either.

Whoo. UV is out of the question. Pricey.

I have read here that lots of plants helps clear the water. If the algae is eating all the "food", how can plants survive?

Can anyone explain why the metal sides of the stock tank are not slimy, but other surfaces in the tank are?

The plants will compete with the algae for the nutrients while adding shade which helps to also control it. One thing I was thinking is that with a 50 gal tank, the water may actually get too hot for the plants, plus you posted that parts of the tank are metal... do you have a thermometer in the pond?

Yes, it is a metal stock tank. It is under a tree and the sun does not hit the metal. The water stays pretty cool. No thermometer.
I added 2 bunches of hornwort and a water lily with a lot of leaves which cover about 1/3 of the surface. No change in the water yet.

Ernie, would it be possible for you to post pics of your pond, it may help us to help you :)

Will try to find my camera!
Ha!

When you find the camera.... take pics of the front, both sides and the top of the pond... and :) The area that the pond resides in... it all helps.