I just realized that I probably have an answer. I don't actually have real filtration at my in pond pump. It resides inside its own case with lots of slits, but that just keeps out fish and big crud. You probably have some fine sponge stuff or something that is too fine for a small area right at the pump inlet. You may not really need any filtering and to the extent you do, it should probably be a larger area of filtering. Perhaps you could make a small 'Skippy' filter (google it). They actually call them 'mini me' filters. It is a diy filter thing in a small garbage can. As I have mentioned many times, my filtering is mostly via my stream with plants whose roots catch everything. You could add a baby streamlet and probably do as well as a small Skippy.

btw, I gather that your pond isn't very deep because it would then have a larger volume. Small volume means less filter need etc.

I was going to suggest the kind of container that chas045 mentioned also. The box itself will also help to block out larger particles while the remaining particles go into the box and get filtered out, along with water.

Otherwise, the external filters work great, too, if you have space for it.

I wouldn't worry about a filter for a pond that size, mine is 3'x5' and I don't have one. The pump is more to provide some aeration and keep the mosquitos from laying eggs on the surface. Currently I have a 12 volt solar panel and a pump that's just under the surface, so it doesn't suck up mud and debris. There is a screen to keep larger material out, but is easy to clean. It only runs when sunny out, but is not really needed in winter when the fish are inactive.

10' x 5' and 350 gallons? 350 gallons is equal to about 47 cubic feet so that would mean the depth of your pond is about 1 foot. That doesn't sound right.

Also, look into airlifts. They are gaining ground in Europe for pond circulation because you can move a lot of water for very little electricity (elect. is very expensive there) compared to pumps. You should have good circulation in order to minimize anaerobic zones that can breed pathogenic bacteria that can make fish sick.

Thanks everyone - your information have been extremely helpful. And Nil13 usda:9a sunset:21 LA,CA (Mount Wash.) you were right - the pond is not 350 it's 750 gallons!