How can a business's community relations department address the most pressing concerns facing communities today?
The answer to this will vary from community to community and from business to business. Different communities will have different "pressing concerns" and different businesses will be more or less able to address those concerns.
The main generic thing that can be done is for businesses to have community relations departments that are well-funded and are really taken seriously by the firm as a whole. The community relations departments need to be on the lookout for ways they can help the community. They should have good connections with various civic groups such as sports leagues or service organizations. Through these connections, they can learn of the things that are most needed in the community. They can then try to use the firm's resources (within reason, of course) to help to address the major concerns felt by the community.