Okay, I have written a lot, so please take your time to read everything through, lets start with one basic thing, the tank is too small and may I ask if you cycled the tank before adding fish?
For now I am going to make the assumption that this is all down to poor water quality and over stocking
Cycling a tank is extremely important and what happens with this, is the tank goes through the Nitrogen cycle while it builds up beneficial bacteria in the gravel and filter. So in a nutshell this is what happens in cycling:
Ammonia levels build up and bacteria develop to turn the ammonia into nitrite, then bacteria develop again to turn the nitrite into nitrate, then water changes are used to keep the nitrate levels safe.
Now ammonia and nitrite are harmful to fish and can cause illness or in some bad cases it can also kill them. Different fish have different sensitivities, so it is possible that your Moor is more sensitive and is affected by poor water quality, whereas the other fish aren't.
The absolute minimum
stocking rules for Fancy Goldfish (including Moors) is that you have 20 gallons for the first fancy goldfish and then 10 gallons for each additional fancy goldfish you add. I would personally increase this and for keeping fancy goldfish, I would say start with 100 litres/26 gallons for the first fish and then add 60 litres/15 gallons for each additional fish, generally the bigger you can go, the better it is for the fish.
So you have too many fish in there as I believe you must have around 13 tetras and the pleco, which with the Moor makes this 20 gallon tank over stocked.
Because of this I would suggest you increase your water changes to at least 50% once a week, this will help with the extra waste produced by the fish (25% a week is suited for tropical fish, goldfish however are extremely messy and require more water to be removed).
I would also advise getting your aquarium water tested, either buy your own test kit (I would recommend a liquid test kit as they are more economical) or get the water tested at a local aquatic store or aquatics department in a pet store.
Basically if there are any ammonia/nitrites in the tank, you need to know and will have to up the water changes. I had nitrites present in my goldfish's tank once and had to perform daily water changes until the levels went back down (so having a test kit at home was very useful because I was able to test the water each day to see if a water change was needed).
The complications with the Moor could also be down to temperature, as I imagine that tank must be pretty warm with the Tetra's in there, the website I've linked below shows the following warning with your current tank (it also mentions the pleco needs driftwood so be sure you have some!);
Warning: Water temperature requirements are not fully compatible between all selected species.
=> 24 - 28C: Ember Tetra
=> 22 - 27C: Albino Bristlenose Pleco
=> 19 - 23C: Black Moor Goldfish
Or the Moors issues could be down to aggression from the other fish due to the over stocking, have you seen the pleco or tetras nipping at the Moors fins?
You will also need to think about what you want to do with regards to the over stocking, maybe check this website about stocking for the tetras and plecos; (Link is not visible to guests. Please register to view.
Also be aware that tetras may get eaten when the moor has grown, so I would suggest thinking about whether your want to keep tropical fish or goldfish, I know some types of pleco can sometimes be kept with goldfish though (what type do you have?).
If you opt for tropical, it's possible that the tetras and pleco (depending on the breed of pleco) would be happy in that 20 gallon tank you have. However if you wish to keep goldfish, you will need to invest in a much larger tank and filtration
Take your time to read through what I've read, as I can appreciate it is a lot to take on board!