I also keep a 10 gallon SW tank. it is not my first (a 3 gallon is my first) but its a TON of work to make sure that everything is perfect. since you have a tiny water volume, even the smallest slip up will usually mean an algae bloom. The larger the water volume, the more leeway you have. I would gladly ditch my smaller 10 gal (i would never ditch my 3 gal sicne thats a novelty lol) and get a 100 gal SW tank just for the ease of mind and maintenance.
I personally am running an AC110 HOB filter on my 10 gal. you may think thats overkill, but flow and additional water volume will really help a small system like 10 gal.
Its perfectly fine if you cant get a bigger HOB filter, in your case i would get a powerhead into that tank, something like a Koralia nano or evo.
Flow is very important, keeping your live rock in top shape is very important.
If you put carbon into your filter while cycling your SW tank, i can almost guarantee you didnt cycle it correctly. (and if you didnt use Live rock, than you certainly did it not do it the way it is meant to be done)
You see carbon will adsorb ammonia directly, not allowing the nitrifying bacteria in your live rock to do their job, thus preventing your tank from cycling.
Carbon is useful in SW though but only after you normally cycle your tank. Its useful in setups with coral, and is optional in FOWLR setups.
Sponges or filter foam should not be used. that type of stuff is ok for FW, and perhaps arguably good enough for SW FOWLR setups, but its a no go if you want to keep sensitive fish and invertebrates, so no foam in reef setups. Reason it that foam, especially if not rinsed frequently enough, will eventually cause a buildup of mad detritis and nitrates.
Filter floss is a better alternative, and just a little bit of it in your filter will do a good job of removing particulate matter from your water. toss the filter floss any time you see it turning brown. usually its good to replace it twice a week, so use small amounts at a time. you can run filter floss when youre cycling.
Right now what i suggest is remove the carbon for now, replace with some filter floss and add some source of ammonia to your tank (hopefully some of your rock is still alive). pure ammonia will work here in SW as well. it is sold in hardware stores as ammonium hydroxide. make sure you dont get the stuff with detergents and soaps. should be pure ammonia and it usually costs much less than most ammonia cleaning products. a big jug will cost around 2-3 dollars.
let this tank run and cycle with ammonia just as you would a FW tank. test your water. you should know the drill of cycling a FW tank. apply your knowledge here now.
