hunting would have meant riding to hounds and chasing foxes or stags.

I don't think JA shot, though some women did.

IMHO, this is just a joke about the amount of birds around that paarticular year. They must have had a dilligent gamekeeper at Godmersham or not many foxes to take the young birds.

It's on a par with Mrs Bennet's daft comment to Bingley in P+P, chapter 53;