You get a clue if you read on a bit: "Queenie begs to taste, and now and again my friend sneaks her a mite, though insisting we deprive ourselves. "We mustn't, Buddy. If we start, we won't stop. And there's scarcely enough as there is."
He is insisting that, in giving the dog a tiny piece of food, we are preventing ourselves from having something that is important to us.
Student or Learner