it would be difficult to fit a sump underneath a 20g half moon tank, but you could conceivably fit one beside the stand in a cabinet. i'd recommend a 20g high sump and go with a cadlights pipeless protein skimmer and a small auto top off system. you can plumb it all together using an eshopps nano 300gph external overflow and a mag drive 3 pump - or a quiet one pump with a similar gph rating. id recommend going with pvc plumbing if at all possible.
keep reading and researching. there are many other options out there - and probably the most simple is to keep the filtration that you have and change out the lights and start doing more water changes. small tanks still need a little love and attention every now and then and a weekly water change might be all that is needed.
when you do a water change, try to remove as much of the detritus out of the system as possible. stir it up with a small powerhead or even a turkey baster so that you can siphon the majority of it out of the system.
if you could, give us a list of your current equipment and maybe we can judge whether you need to go through all of that trouble or not.