Gonna cost you between $100 and $200 to fix this. You need a treadle feeder and the good news is that even the poorly designed Chinese made feeders like the Grandpa feeder, the Rent a coop feeder, or one of the other myriad of clones will stop small birds. Not pigeons, the Grandpa feeder and its clones are NOT pigeon proof due to that wide step, two pigeons can trip the treadle. If you are a prime member, no shipping, you can get one of these Chinese made feeders for under $100 or $179 for the Grandpa or Rent a coop feeder. An American made feeder starts at $85 plus shipping. Shipping runs from $18.00 for a central U.S. state up to $35.00 if you live on one of the two coasts.
So you will pay a little extra for an American made feeder but it is a must more modern design that is not only bird proof, it is pigeon proof, rat proof, and with some caveats it is squirrel proof. That being if a squirrel gets past the heavy spring loaded door it will trap itself and you have to dispose of the squirrel. And you get actual customer service and all the repair parts you might need. Even the sheet metal, bar stock, and packaging is all American made, some of the hardware like nuts and bolts might be foreign made.
There are cheaper treadle feeders, we have one for $36.00 and the aluminum (not really safe for feed) ones will run $60.00 with "free" shipping on
Amazon or
Ebay. But they hold only 12 pounds, only four pounds if you want the hens not to rake feed out of it, and they can be overwhelmed by wild birds if they swarm which they will do.
Before you spend hard earned money though check your feed loss. Figure a quarter pound of feed per day per hen, ignore roosters unless you have a lot of them. Figure out what it is costing you and calculate the payback on a feeder. Might not be worth it financially if it is just a few wild birds. Might pay for the feeder in a few months. If close, consider the amount of pests and disease coming into the flock. Sounds like you already are paying a high cost with hen loss.