Central banks care about inflation, but not about the inflation people care about. Because energy prices are volatile, they are not included in the price index a central bank typically looks at, because including it would made the indicator less informative. At they also include food prices, because those fluctuate a lot as well, in particular because of seasons. That can make sense for a rich country, where food represents only a small portion of the budget. For poorer countries, excluding food is more controversial.
Luis Catão and Roberto Chang note that world food prices seem to cause worldwide inflation, and this should have implications for inflation in small open economies that take food prices as given. The question is then whether central banks should react to such terms of trade shocks. For a net food importer, Catão and Chang find that including food in the price index relevant to the central bank is welfare enhancing. The reason is that if food has a larger weight domestically than in the rest of the world, the real exchange rate and the terms of trade can move in opposite directions following a food price shocks. The welfare improvement comes from a change in the correlations in aggregates leading to smoother consumption, but it possibly results in higher inflation and higher volatility of output and employment. It is thus not obvious including food prices would then be an easy sell.