Yes, this can be done. There are two types of polygon barriers: restricted polygon barriers and scaled polygon barriers. When you create your polygon barrier, you can open its properties (right-click the barrier in the Network Analyst Window and click 'Properties...') and see its associated field values. Make sure the 'BarrierType' field value is set to 'Scaled Cost'. Then, assuming you are solving on cost attribute X, set the 'Attr_X' field value to some positive value less than 1 (e.g., 0.25). This field value sets the scale factor applied to cost attribute X for all underlying network elements covered by the polygon. Setting this value less than 1 effectively causes the solvers to "prefer" these areas more than they would have normally. You can adjust the scale factor depending on how much you wish to prefer these polygon areas.
