Can psychiatrists also treat chronic pain?

Yes. Pain affects all aspects of one's life. In chronic situations, pain may have devastating effect on a patient's emotional well-being. A psychiatrist or a psychologist can help patient deal much more effectively with depression, anxiety, etc. They can also help with improving patient's coping skills.
Yes. Some psychiatrists do treat pain. They can use cognitive and behavioural therapy as well as medications. They can't use injections however.