The honda reserve system is a sliding shaft with a straw that pull out of the tank and various levels from inside a metal screen. Sounds complicated but think of it this way, it pulls on normal untill there is 2"s or so of fuel left, swap the valve and it lowers the tube so that the last few inches of gas can be sucked up. That trash is plugging the screen that is a verticle shaft on a honda so when the fuel is lower is pulling from a partial amount of the screen at some point and that restricts the flow. I have never worked on a polaris so im not sure how they work.
Hondas carbs are a bear to get to!!!
The honda kits i think are like $30 to rebuild and they take 15 mins once you figure out how to pull the carb and contort your hands. The first time it toook me 1 hour to pull it, i can now do it in 15-20 mins or less. Just what has to come off and how it does is what took me so long to learn. Getting your air or fuel screw (depending upon which you have) is what will take the most finess. I can buy a brand new chineese carb for my 1989 honda 300trx for right around $100. I had to replace my fuel petcock as it was leaking, got a chineese replacement of ebay for $20 or less if i remember right. Ebay is your friend for cheap ATV parts.
But that trash you see in the tank SHOULD not be making it to the carb, maybe really fine pieces, i mean really fine could be in there, but most of that stuff will stay in the tank forever.
If your not running 100% gas and running any amont of Ethonol content I WOULD cut the fuel off and let it run dry. That ethonol leaves funny deposits in carbs that resemble sand and will plug orfaces and stick needles to the seats. Trust me i have seen in in several things.
That said i will only put real gas in my 4wheeler and do not cut the fuel off unless i think about it when i am trailering it so that it will not flow back into the cylinder when it goes over bumps or if the float gets stuck while its on the trailer.
My atv does not flood out while running or sitting. Did when i got it though.