Focusing on core inflation might increase the likelihood that monetary policy would respond to underlying inflation developments that promised to be persistent, and not on transitory changes in inflation. Nevertheless, most inflation-targeting regimes set their explicit inflation target in terms of overall inflation.
As long as policy is focused on achieving a longer-run inflation outcome, as opposed to meeting the target each quarter or even each year, a target for overall inflation can work well, because swings in energy and food prices that lead to shorter-run inflation blips tend to dissipate or wash out over longer periods.