What do you have over that farm gate?
My birds are more likely to go under the dog fence, so if you have several inches under a section of the chain link, fill that in with something. I put some 4x4" beams at my low spots, and it's helped a lot. The gap between your fence and chainlink gate isn't helpful either.
That's a very short fence! Consider attaching poles and higher wire, maybe even cheap chicken wire, to make the thing taller and discourage birds sitting on the top of the fence.
Then, there's electric poultry fencing for outside your dog fence.
My dogs love chicken too, and keeping them apart is necessary here also.
Mary