Hi boogah and welcome to the forum, (firstly let me warn you this is rather long, so please take your time)
What type of filter do you have? Is it an internal filter or external filter?
It can be possible to cover the inlet on some filters, to lower the current, certain sponges can be used or covering the inlet with a material such as the foot end of a tight, these tactics are used in shrimp tanks and fry tanks, to stop them from being sucked in to the filter, so could help out with your situation.
How badly damaged are the fins? Could you describe it in further detail? Do you have a filter in the hospital tank?
Sometimes depending on the damage, fish can heal themselves as long as the water quality is good. You will want to keep an eye on things and make sure no secondary infections occur, such as fin rot.
When I say 'good water quality', you may wonder what that means, so I'll explain briefly.
In an aquarium there are 3 main elements that have to be considered; Ammonia, Nitrite and Nitrate.
When you first set up a tank, there is a process called 'cycling', simply where bacteria develops in the tank and filter, so that the harmful substances of ammonia is converted to nitrite and then more develop to convert the nitrite into nitrate.
So in a cycling tank you see ammonia levels rise, and then begin to fall, followed by nitrite levels rising and then beginning to fall, which is lastly followed by nitrate levels which begin to rise.
Ammonia and Nitrite are extremely bad news for fish and can cause illness as well as death, if not taken care of.
Nitrates however aren't as bad, as long as the concentration is below 40ppm. Weekly water changes, keep's the nitrates under control.
So as fish keepers, we use test kits to monitor these elements and re act accordingly. Ideally in a fully cycled system Ammonia and Nitrites will be 0ppm and Nitrates will be below 40ppm.
As Fantailer has said, tropical fish -depending on what they are- generally aren't compatible. Often tropical fish have different requirements in terms of PH and temperature, do you know what breed of fish they are? That would be helpful
Also when you describe your other fish as 'sucker cleaning fish' I am going to assume you mean a pleco of some kind. Again knowing the breed would be helpful, sometimes pleco's can be kept with goldfish, but there have been cases of the pleco sucking the slime coat from a goldfish. I'm not sure how common this situation can be, but it's something to be aware of. And also depending on the breed, will affect it's suitability for your 10 gallon tank, as some pleco's can grow to huge sizes.
May I also ask what type of goldfish the other two are?
Are they fantails or another fancy goldfish type, OR are they single tailed such as a common goldfish or shubunkin?
If you're not sure, you could always look things up on google images
In general for goldfish (as fantailer has already said) they grow pretty big and are extremely messy fish, excreting a lot of waste through poop and their gills give some off as well. As a result they need a lot of tank space.
The minimum stocking rule for fancy goldfish types, is 20 gallons for the first fish and then 10 gallons for each additional fish. So assuming your other two goldfish are fancy, you would need a 40 gallon tank, minimum. To successfully keep the goldfish.
If the other two goldfish are single tailed, then it's an entirely different situation.
As single tailed and fancy goldfish should never be mixed, as the single tailed fish are much quicker and can out compete the fancy varieties for food as well as potentially stressing them out, by being such agile and quick movers.
As a result of their active behaviour, single tailed fish are more suited to large aquaria (with a length being a minimum of 6 foot) or even better, a decent sized pond of around 1000 litres or more.
So it all depends on what types of fish you have
If you have any further questions, please feel free to ask, how is the Moor doing?
Is he eating? Are there any more symptoms physically you can see asides from the fins? Or is he acting oddly, such as bottom sitting?