Birds are very messy eaters! I have to admit that I had the same problem as you, and eventually moved my feeding pole to near the patio. I have put a large shallow pot of soil underneath the main feeder so hopefully most of the fallen seeds fall into it. That way the ground feeders - blackbirds, robins etc. can get the leftovers! And I can weed the bowl rather than the grass!
Alternatively, if you are handy at DIY, make a wooden frame, with legs, and cover it with mesh. This will catch many of the seeds, allow drainage so they don't go mushy, and allow light to get to your grass underneath. It's also easy to move when you want to cut the grass!
