To be filled with a deeply unpleasant emotion, such as grief, remorse, dejection, etc. I am sick at heart to think that I may have contributed to her death in some way. Poor Tom's been sick at heart ever since his mother died.

To feel a deeply unpleasant emotion, such as grief, remorse, dejection, etc. I feel sick at heart to think that I may have contributed to her death in some way. Poor Tom's felt sick at heart ever since his mother died.