Perhaps try food blocks that contain mostly safflower seed or thistle. Other than that, I haven't seen any cages for food blocks, just one that is made for a ground feeder.
If the woodpeckers, chickadees, etc., come in the evening and know the food will be out there at that time, I don't think they will stop coming around.
Starlings and Grackles are hungry birds. I just put food out for all the birds, since I've given up with trying various feeders. The little birds and cardinals get somewhat intimidated, but they come back. Cardinals like safflower seed, too, and I've seen one stand his ground with a Grackle. I don't think Starlings or Grackles are attracted to safflower or the thistle seed that finches prefer.
The Red-Bellied woodpeckers don't get bullied, either.