If you move the fish to a new tank & not the filter, then you have lost your bio-filter & need to cycle the new tank. 1 week is not long enough to cycle the tank. The bacterial bloom is indicative of an ammonia spike, you are prob in the start of the nitrite phase. If you still have your old tank running, you can shorten the cycle by seeding the new tank with the filter media from the old.
The whole process is extremely stressful to the fish, hence the problems you are seeing. From the picture I don't think this is fungus, possibly a bacterial infection (eg flexobactor aka columnaris), but more than likely an ammonia burn (possibly infected).
1st order of treatment is clean water. Until you remove the ammonia, the wounds will not heal. It is important to test the water to see where you are at. In the absence of test kit, I would do a 50% pwc on spec, and do that daily until you can test the water.
What other fish or plants you have? If it is just goldfish, I would also suggest adding salt. <Aquarium salt, or kosher or sea salt.> Salt will be effective against a host of external infections. In addition, it will also protect against nitrite poisoning - which is the next thing to watch for in a cycling tank. <Nitrtie binds to hemoglobin, preventing uptake of oxygen. The fish will act like it has no O2 - rapid breathing, gulping air, then lethargy.> Assuming there is no salt sensitive creatures in the tank, I would suggest adding salt to 0.1%. For a 55 gal, that would be ~200 g or ~12 tablespoons of salt. You want to predissolve the salt in water, and you might want to add 1/2 now & 1/2 the next day. When doing the pwc, you would need to maintain salt level by adding the same amount of salt to your change water. <So if you are changing out 25 gals of water, add 6 tablespoons of salt to the new water.>
This is relatively low level of salt, but is usu. effective for minor surface wounds & infections. you can up that to .3% if you are treating ich or other major infections. If the fish is responding, I would keep the salt level up for a week or 2 (all the time maintaining good water conditions), then let the level drop with pwc. It might be beneficial to maintain low level of salt (~0.01%) until your cycle is completed.
This is more detailed instructions on using salt, although the level is to 0.3% for ich. <You can 1/2 or 1/3 the dose for now.>
Treating Ich with Salt