Something that is incessant continues without interruption—and is more often than not a source of irritation! Incessant chatter near your workstation might disrupt your concentration; incessant whining in any circumstance might deplete your patience; and incessant demands, as from a boss or from any other source, might make you feel exhausted or taken for granted (or both!). When not used to describe undesirable situations such as these, you may find incessant used in a more neutral sense to describe continuous and unrelenting rainfall, or other weather events that carry on seemingly to no end.
Something that is stale has lost freshness, such as a stale slice of bread, or has lost novelty or interest, such as a stale joke. That crusty old joke may also be described as hackneyed. Something that is hackneyed is commonplace or trite and worn out by overuse. This term is especially used of words, remarks, or styles of expression that are clichéd to the point of seeming lifeless and uninteresting. So synonym seekers beware: while it may be tempting to slather this adjective on a piece of stale bread, hackneyed simply doesn't make sense as a descriptor for brittle day-old baked goods!
To allay something is to put it to rest or to quiet it. This sense of the word is mostly used to talk about laying to rest fear, doubt, suspicion, or anger, possibly by making the emotion seem unjustified. Allay is also used to talk about lessening or mitigating something, such as pain. Soothe is similar (an ointment can soothe sunburned skin, for instance). But a key difference is that soothe is sometimes used to talk about people (rather than their concerns or fears)—and in these cases, it may suggest tranquility and even comfort. A person might be soothed or comforted by a warm cup of tea, for instance, but not allayed.