Sharpen and hone refer to increasing the fineness of a point or thinness of a cutting edge on a tool to increase its effectiveness. Sharpen is the everyday word we apply to knives, blades, edges, pencils, and claws. It’s also commonly extended to mean improving an ability or skill. A challenge that sharpens your skills improves them, so that you can do a better job with them. Now, something that needs sharpening could be dull, whereas hone often suggests making something sharp even sharper. A hone is a whetstone (sharpening stone) of particularly fine, closely-packed grain. The verb has a literal definition (honed my carving knife), but most often refers to refining, polishing, and perfecting skills, techniques, or abilities, as in difficult assignments that honed our critical thinking skills.
Heave ho! Discard and jettison both refer to getting rid of things that we don’t want or need. When it’s not a card game move, discard usually refers to throwing something away, as when a recipe instructs us to remove the stems and discard. Jettison is a strong synonym for discard, especially when describing getting rid of things that have value for the sake of making progress on something. Literally, jettison means to cast cargo overboard in order to lighten the load of a ship (and subsequently an air- or spacecraft): They jettisoned the lunar module to prepare for descent. By extension it also can mean to offload any type of obstacle that is impeding success, including ideas and practices. As a manager, you may have to jettison the notion that you will be able to please everyone.
Promising and auspicious are used to describe a thing or person with qualities that lead us to expect they will turn out favorably. A promising first draft possesses elements that, if developed, will constitute a good paper, and people have high hopes for a promising young man or woman. The optimal time to be promising is, of course, at the start (a vacation off to a promising start). Auspicious start is also common usage, and can be used synonymously. Strictly, however, auspicious means “boding well,” as if foretold by an omen. An auspicious sign bodes well for something or someone. Auspicious is frequently used to describe times: an auspicious occasion, day, or moment. Here it suggests that the outcome of something is lucky, well-timed, or favorable at that time. In usage, the meaning of auspicious varies between "promising" and “favorable for.”