The noun piece can mean a portion, part, or unit of something and has numerous definitions, giving it a broad range of application: a piece of cake, a piece of land, a piece of furniture, a piece of music, a piece of pottery. One of its definitions overlaps with the core meaning of the synonym fragment: a part that is broken off or detached (scattered fragments of the broken vase). By itself, piece is more general and ambiguous: a piece of pottery is more likely to mean an individual bowl or mug than a broken-off piece, whereas there’s no mistaking what a fragment of pottery refers to. While piece and fragment both name things that are less than whole, the connotations of fragment are more particular: fragment suggests a broken, inconsequential, incomplete part, with irregular or imprecise outlines or boundaries.
Mushy is an informal word meaning overly emotional or sentimental (mushy love letters). The more elevated adjective maudlin means tearfully or weakly emotional, or foolishly sentimental (a maudlin story of a little orphan and her lost dog). Maudlin emotion is excessive because it’s disproportionate to what motivates it. If a TV commercial for soap makes you sob, you are being maudlin, although the same could be said of the commercial, with its effective but shallow and insincere formula for eliciting tears. As for how this adjective came to suggest preternaturally hyperactive tear ducts: before English speakers were describing commercials and sad stories as maudlin, they were using the term as a noun to refer to Mary Magdalene, who was traditionally depicted weeping.
To carry something is to take or support it from one place to another. This very common verb does not, on its own, suggest difficulty or a high degree of effort; it merely states a mode of conveyance: I carried the casserole to the neighbor’s house. The synonym lug, on the other hand, suggests strenuous effort. This verb is used to talk about pulling or carrying heavy or ponderous things with force or great exertion. If you lug a suitcase up a flight of stairs, the suggestion is that you are transporting it, maybe even dragging it, with some difficulty. It’s no surprise that this verb is sometimes chosen to emphasize or even exaggerate the burdensome nature of a task.